Invariant Theory Nq

Invariant theory nqFrom Wikipedia, the free encyclopedia

Contents

1 Newtons identities 11.1 Mathematical statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Formulation in terms of symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Application to the roots of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Application to the characteristic polynomial of a matrix . . . . . . . . . . . . . . . . . . . 31.1.4 Relation with Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 A variant using complete homogeneous symmetric polynomials . . . . . . . . . . . . . . . 41.2.2 Expressing elementary symmetric polynomials in terms of power sums . . . . . . . . . . . 41.2.3 Expressing complete homogeneous symmetric polynomials in terms of power sums . . . . 41.2.4 Expressing power sums in terms of elementary symmetric polynomials . . . . . . . . . . . 51.2.5 Expressing power sums in terms of complete homogeneous symmetric polynomials . . . . 51.2.6 Expressions as determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Derivation of the identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 From the special case n = k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Comparing coefficients in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 As a telescopic sum of symmetric function identities . . . . . . . . . . . . . . . . . . . . 9

1.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Nullform 112.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Osculant 123.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Perpetuant 134.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Polynomial ring 155.1 The polynomial ring K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.1.2 Degree of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

i

ii CONTENTS

5.1.3 Properties of K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.1.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 Polynomial evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 The polynomial ring in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3.2 The polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.3 Hilberts Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.4 Properties of the ring extension R R[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5.1 Infinitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5.2 Generalized exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5.4 Noncommutative polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.5.5 Differential and skew-polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Quantum invariant 246.1 List of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Quaternary cubic 277.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Sylvester pentahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Quippian 298.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 1

Newtons identities

In mathematics,Newtons identities, also known as theNewtonGirard formulae, give relations between two typesof symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at theroots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P(counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identitieswere found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They haveapplications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics,as well as further applications outside mathematics, including general relativity.

1.1 Mathematical statement

1.1.1 Formulation in terms of symmetric polynomials

Let x1, , xn be variables, denote for k 1 by pk(x1, , xn) the k-th power sum:

pk(x1, . . . , xn) =n

i=1xki = x

k1 + + xkn,

and for k 0 denote by ek(x1, , xn) the elementary symmetric polynomial (that is, the sum of all distinct productsof k distinct variables), so

e0(x1, . . . , xn) = 1,

e1(x1, . . . , xn) = x1 + x2 + + xn,e2(x1, . . . , xn) =

1i n.

Then Newtons identities can be stated as

kek(x1, . . . , xn) =k

i=1

(1)i1eki(x1, . . . , xn)pi(x1, . . . , xn),

valid for all n 1 and k 1.Also, one has

0 =k

i=kn

(1)i1eki(x1, . . . , xn)pi(x1, . . . , xn),

1
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2 CHAPTER 1. NEWTONS IDENTITIES

for all k > n 1.Concretely, one gets for the first few values of k:

e1(x1, . . . , xn) = p1(x1, . . . , xn),

2e2(x1, . . . , xn) = e1(x1, . . . , xn)p1(x1, . . . , xn) p2(x1, . . . , xn),3e3(x1, . . . , xn) = e2(x1, . . . , xn)p1(x1, . . . , xn) e1(x1, . . . , xn)p2(x1, . . . , xn) + p3(x1, . . . , xn).

The form and validity of these equations do not depend on the number n of variables (although the point where theleft-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities inthe ring of symmetric functions. In that ring one has

e1 = p1,

2e2 = e1p1 p2,3e3 = e2p1 e1p2 + p3,4e4 = e3p1 e2p2 + e1p3 p4,

and so on; here the left-hand sides never become zero. These equations allow to recursively express the ei in terms ofthe pk; to be able to do the inverse, one may rewrite them as

p1 = e1,

p2 = e1p1 2e2,p3 = e1p2 e2p1 + 3e3,p4 = e1p3 e2p2 + e3p1 4e4,

...

In general, we have

pk(x1, . . . , xn) = (1)k1kek(x1, . . . , xn) +k1i=1

(1)k1+ieki(x1, . . . , xn)pi(x1, . . . , xn),

valid for all n 1 and k 1.Also, one has

pk(x1, . . . , xn) =k1

i=kn

(1)k1+ieki(x1, . . . , xn)pi(x1, . . . , xn),

for all k > n 1.

1.1.2 Application to the roots of a polynomial

The polynomial with roots xi may be expanded as

ni=1

(x xi) =n

k=0

(1)n+kenkxk,

where the coefficients ek(x1, . . . , xn) are the symmetric polynomials defined above. Given the power sums of theroots
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1.1. MATHEMATICAL STATEMENT 3

pk(x1, . . . , xn) =n

i=1

xki ,

the coefficients of the polynomial with roots x1, . . . , xn may be expressed recursively in terms of the power sums as

e0 = 1,

e1 = p1,

e2 =1

2(e1p1 p2),

e3 =1

3(e2p1 e1p2 + p3),

e4 =1

4(e3p1 e2p2 + e1p3 p4),

...

Formulating polynomial this way is useful in using the method of Delves and Lyness[1] to find the zeros of an analyticfunction.

1.1.3 Application to the characteristic polynomial of a matrix

When the polynomial above is the characteristic polynomial of a matrix A (in particular when A is the companionmatrix of the polynomial), the roots xi are the eigenvalues of the matrix, counted with their algebraic multiplicity.For any positive integer k, the matrix Ak has as eigenvalues the powers xik, and each eigenvalue xi of A contributesits multiplicity to that of the eigenvalue xik of Ak. Then the coefficients of the characteristic polynomial of Ak aregiven by the elementary symmetric polynomials in those powers xik. In particular, the sum of the xik, which is thek-th power sum sk of the roots of the characteristic polynomial of A, is given by its trace:

sk = tr(Ak) .

The Newton identities now relate the traces of the powers Ak to the coefficients of the characteristic polynomial ofA. Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can beused to find the characteristic polynomial by computing only the powers Ak and their traces.This computation requires computing the traces of matrix powers Ak and solving a triangular system of equations.Both can be done in complexity class NC (solving a triangular system can be done by divide-and-conquer). Therefore,characteristic polynomial of a matrix can be computed in NC. By the Cayley-Hamilton theorem, every matrix satisfiesits characteristic polynomial, and a simple transformation allows to find the matrix inverse in NC.Rearranging the computations into an efficient form leads to the Fadeev-Leverrier algorithm (1840), a fast parallelimplementation of it is due to L. Csanky (1976). Its disadvantage is that it requires division by integers, so in generalthe field should have characteristic, 0.

1.1.4 Relation with Galois theory

For a given n, the elementary symmetric polynomials ek(x1,,xn) for k = 1,, n form an algebraic basis for the spaceof symmetric polynomials in x1,. xn: every polynomial expression in the xi that is invariant under all permutations ofthose variables is given by a polynomial expression in those elementary symmetric polynomials, and this expressionis unique up to equivalence of polynomial expressions. This is a general fact known as the fundamental theoremof symmetric polynomials, and Newtons identities provide explicit formulae in the case of power sum symmetricpolynomials. Applied to the monic polynomial tn +

nk=1(1)kaktnk with all coefficients ak considered as free

parameters, this means that every symmetric polynomial expression S(x1,,xn) in its roots can be expressed insteadas a polynomial expression P(a1,,an) in terms of its coefficients only, in other words without requiring knowledgeof the roots. This fact also follows from general considerations in Galois theory (one views the ak as elements of a
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base field with roots in an extension field whose Galois group permutes them according to the full symmetric group,and the field fixed under all elements of the Galois group is the base field).The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sumsymmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact thefirst n power sums also form an algebraic basis for the space of symmetric polynomials.

1.2 Related identities

There are a number of (families of) identities that, while they should be distinguished from Newtons identities, arevery closely related to them.

1.2.1 A variant using complete homogeneous symmetric polynomials

Denoting by hk the complete homogeneous symmetric polynomial that is the sum of all monomials of degree k,the power sum polynomials also satisfy identities similar to Newtons identities, but not involving any minus signs.Expressed as identities of in the ring of symmetric functions, they read

khk =k

i=1

hkipi,

valid for all n k 1. Contrary to Newtons identities, the left-hand sides do not become zero for large k, and theright-hand sides contain ever more non-zero terms. For the first few values of k, one has

h1 = p1,

2h2 = h1p1 + p2,

3h3 = h2p1 + h1p2 + p3.

These relations can be justified by an argument analogous to the one by comparing coefficients in power series givenabove, based in this case on the generating function identity

k=0

hk(X1, . . . , Xn)tk =

ni=1

1

1Xit.

Proofs of Newtons identities, like these given below, cannot be easily adapted to prove these variants of those iden-tities.

1.2.2 Expressing elementary symmetric polynomials in terms of power sums

As mentioned, Newtons identities can be used to recursively express elementary symmetric polynomials in termsof power sums. Doing so requires the introduction of integer denominators, so it can be done in the ring Q ofsymmetric functions with rational coefficients:and so forth. Applied to a monic polynomial, these formulae express the coefficients in terms of the power sums ofthe roots: replace each ei by ai and each pk by sk.

1.2.3 Expressing complete homogeneous symmetric polynomials in terms of power sums

The analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, givingequationsand so forth, in which there are only plus signs. These expressions correspond exactly to the cycle index polynomialsof the symmetric groups, if one interprets the power sums pi as indeterminates: the coefficient in the expression for hk
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1.2. RELATED IDENTITIES 5

of any monomial p1m1p2m2plml is equal to the fraction of all permutations of k that have m1 fixed points, m2 cyclesof length 2,, andml cycles of length l. Explicitly, this coefficient can be written as 1/N whereN = li=1(mi! imi); thisN is the number permutations commuting with any given permutation of the given cycle type. The expressionsfor the elementary symmetric functions have coefficients with the same absolute value, but a sign equal to the sign of, namely (1)m2+m4+....It can be proved by considering the following inductive step:

mf(m;m1, ...,mn) = f(m 1;m1 1, ...,mn) + ...+ f(m n;m1, ...,mn 1)

m1

ni=1

1

imimi!+ ...+ nmn

ni=1

1

imimi!= m

ni=1

1

imimi!

1.2.4 Expressing power sums in terms of elementary symmetric polynomials

One may also use Newtons identities to express power sums in terms of symmetric polynomials, which does notintroduce denominators:

p1 = e1,

p2 = e21 2e2,

p3 = e31 3e2e1 + 3e3,

p4 = e41 4e2e21 + 4e3e1 + 2e22 4e4,

p5 = e51 5e2e31 + 5e3e21 + 5e22e1 5e4e1 5e3e2 + 5e5,

p6 = e61 6e2e41 + 6e3e31 + 9e22e21 6e4e21 12e3e2e1 + 6e5e1 2e32 + 3e23 + 6e4e2 6e6.

The first four formulas were obtained by Albert Girard in 1629 (thus before Newton).[2]

The general formula (for all positive integers m and n) is:

pm =

r1+2r2++nrn=mr10,...,rn0

(1)mm(r1 + r2 + + rn 1)!r1!r2! rn!

ni=1

(ei)ri

which can be proved by considering the following inductive step:

f(m; r1, , rn) = f(m 1; r1 1, , rn) + + f(m n; r1, , rn 1)

=1

(r1 1)! rn!(m 1)(r1 + + rn 2)! + +

1

r1! (rn 1)!(m n)(r1 + + rn 2)!

=1

r1! rn![r1(m 1) + + rn(m n)] [r1 + + rn 2]!

=1

r1! rn![m(r1 + + rn)m] [r1 + + rn 2]!

=m(r1 + + rn 1)!

r1! rn!

1.2.5 Expressing power sums in terms of complete homogeneous symmetric polynomials

Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to ex-press power sums in term of them:
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p1 = +h1,

p2 = h21 + 2h2,p3 = +h

31 3h2h1 + 3h3,

p4 = h41 + 4h2h21 4h3h1 2h22 + 4h4,p5 = +h

51 5h2h31 + 5h22h1 + 5h3h21 5h3h2 5h4h1 + 5h5,

p6 = h61 + 6h2h41 9h22h21 6h3h31 + 2h32 + 12h3h2h1 + 6h4h21 3h23 6h4h2 6h1h5 + 6h6,

and so on. Apart from the replacement of each ei by the corresponding hi, the only change with respect to the previousfamily of identities is in the signs of the terms, which in this case depend just on the number of factors present: thesign of the monomial li=1hmii is (1)m1+m2+m3+. In particular the above description of the absolute value of thecoefficients applies here as well.The general formula (for all positive integers m and n) is:

pm =

m1+2m2++nmn=mm10,...,mn0

m(r1 + r2 + + rn 1)!r1!r2! rn!

ni=1

(hi)ri

1.2.6 Expressions as determinants

One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first n ofNewtons identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which theelementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramersrule to find the solution for the final unknown. For instance taking Newtons identities in the form

e1 = 1p1,

2e2 = e1p1 1p2,3e3 = e2p1 e1p2 + 1p3,

...nen = en1p1 en2p2 + + (1)ne1pn1 + (1)n1pn

we consider p1 , p2 , p3 , , (1)npn1 and pn as unknowns, and solve for the final one, giving

pn =

1 0 e1e1 1 0 2e2e2 e1 1 3e3... . . . . . .

...en1 e2 e1 nen

1 0 e1 1 0 e2 e1 1... . . . . . .

en1 e2 e1 (1)n1

1

=1

(1)n1

1 0 e1e1 1 0 2e2e2 e1 1 3e3... . . . . . .

...en1 e2 e1 nen

=

e1 1 0 2e2 e1 1 0 3e3 e2 e1 1... . . . . . .

nen en1 e1

.
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1.3. DERIVATION OF THE IDENTITIES 7

Solving for en instead of for pn is similar, as the analogous computations for the complete homogeneous symmetricpolynomials; in each case the details are slightly messier than the final results, which are (Macdonald 1979, p. 20):

en =1

n!

p1 1 0 p2 p1 2 0 ... . . . . . .

pn1 pn2 p1 n 1pn pn1 p2 p1

pn = (1)n1

h1 1 0 2h2 h1 1 0 3h3 h2 h1 1... . . . . . .

nhn hn1 h1

hn =

1

n!

p1 1 0 p2 p1 2 0 ... . . . . . .

pn1 pn2 p1 1 npn pn1 p2 p1

.

Note that the use of determinants makes that the formula for hn has additional minus signs compared to the one foren , while the situation for the expanded form given earlier is opposite. As remarked in (Littlewood 1950, p. 84) onecan alternatively obtain the formula for hn by taking the permanent of the matrix for en instead of the determinant,and more generally an expression for any Schur polynomial can be obtained by taking the corresponding immanantof this matrix.

1.3 Derivation of the identities

Each of Newtons identities can easily be checked by elementary algebra; however, their validity in general needs aproof. Here are some possible derivations.

1.3.1 From the special case n = k

One can obtain the k-th Newton identity in k variables by substitution into

ki=1

(t xi) =k

i=0

(1)kieki(x1, . . . , xk)ti

as follows. Substituting xj for t gives

0 =k

i=0

(1)kieki(x1, . . . , xk)xji for1 j k

Summing over all j gives

0 = (1)kkek(x1, . . . , xk) +k

i=1

(1)kieki(x1, . . . , xk)pi(x1, . . . , xk),

where the terms for i = 0 were taken out of the sum because p0 is (usually) not defined. This equation immediatelygives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) of
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degree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities inn < k variables can be deduced by setting k n variables to zero. The k-th Newton identity in n > k variables containsmore terms on both sides of the equation than the one in k variables, but its validity will be assured if the coefficientsof any monomial match. Because no individual monomial involves more than k of the variables, the monomial willsurvive the substitution of zero for some set of n k (other) variables, after which the equality of coefficients is onethat arises in the k-th Newton identity in k (suitably chosen) variables.

1.3.2 Comparing coefficients in series

Another derivation can be obtained by computations in the ring of formal power series R[[t]], where R is Z[x1,,xn], the ring of polynomials in n variables x1,, xn over the integers.Starting again from the basic relation

ni=1

(t xi) =n

k=0

(1)kaktnk

and reversing the polynomials by substituting 1/t for t and then multiplying both sides by tn to remove negativepowers of t, gives

ni=1

(1 xit) =n

k=0

(1)kaktk.

(the above computation should be performed in the field of fractions of R[[t]]; alternatively, the identity can beobtained simply by evaluating the product on the left side)Swapping sides and expressing the ai as the elementary symmetric polynomials they stand for gives the identity

nk=0

(1)kek(x1, . . . , xn)tk =n

i=1

(1 xit).

One formally differentiates both sides with respect to t, and then (for convenience) multiplies by t, to obtain

nk=0

(1)kkek(x1, . . . , xn)tk = tn

i=1

[(xi)

j =i

(1 xjt)]

=

(n

i=1

xit

1 xit

)nj=1

(1 xjt)

=

ni=1

j=1

(xit)j

[ n=0

(1)e(x1, . . . , xn)t]

=

j=1

pj(x1, . . . , xn)tj

[ n=0

(1)1e(x1, . . . , xn)t],

where the polynomial on the right hand side was first rewritten as a rational function in order to be able to factor outa product out of the summation, then the fraction in the summand was developed as a series in t, using the formula

X

1X= X +X2 +X3 +X4 +X5 +

and finally the coefficient of each t j was collected, giving a power sum. (The series in t is a formal power series, butmay alternatively be thought of as a series expansion for t sufficiently close to 0, for those more comfortable with that;
https://en.wikipedia.org/wiki/Formal_power_serieshttps://en.wikipedia.org/wiki/Polynomial_ring#The_polynomial_ring_in_several_variableshttps://en.wikipedia.org/wiki/Field_of_fractionshttps://en.wikipedia.org/wiki/Formal_derivativehttps://en.wikipedia.org/wiki/Rational_function

1.4. SEE ALSO 9

in fact one is not interested in the function here, but only in the coefficients of the series.) Comparing coefficients oftk on both sides one obtains

(1)kkek(x1, . . . , xn) =k

j=1

(1)kj1pj(x1, . . . , xn)ekj(x1, . . . , xn),

which gives the k-th Newton identity.

1.3.3 As a telescopic sum of symmetric function identities

The following derivation, given essentially in (Mead, 1992), is formulated in the ring of symmetric functions forclarity (all identities are independent of the number of variables). Fix some k > 0, and define the symmetric functionr(i) for 2 i k as the sum of all distinct monomials of degree k obtained by multiplying one variable raised tothe power i with k i distinct other variables (this is the monomial symmetric function m where is a hook shape(i,1,1,1)). In particular r(k) = pk; for r(1) the description would amount to that of ek, but this case was excludedsince here monomials no longer have any distinguished variable. All products pieki can be expressed in terms of ther(j) with the first and last case being somewhat special. One has

pieki = r(i) + r(i+ 1) for1 < i < k

since each product of terms on the left involving distinct variables contributes to r(i), while those where the variablefrom pi already occurs among the variables of the term from eki contributes to r(i + 1), and all terms on the right areso obtained exactly once. For i = k one multiplies by e0 = 1, giving trivially

pke0 = pk = r(k)

Finally the product p1ek for i = 1 gives contributions to r(i + 1) = r(2) like for other values i < k, but the remainingcontributions produce k times each monomial of ek, since any one of the variables may come from the factor p1; thus

p1ek1 = kek + r(2)

The k-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of theform r(i) cancel out.

1.4 See also Power sum symmetric polynomial

Elementary symmetric polynomial

Symmetric function

Fluid solutions, an article giving an application of Newtons identities to computing the characteristic polyno-mial of the Einstein tensor in the case of a perfect fluid, and similar articles on other types of exact solutionsin general relativity.

1.5 References[1] Delves, L. M. (1967). A Numerical Method of Locating the Zeros of an Analytic Function. Mathematics of Computation

21: 543560. doi:10.2307/2004999.

[2] Tignol, Jean-Pierre (2004). Galois theory of algebraic equations (Reprinted. ed.). River Edge, NJ: World Scientific. pp.3738. ISBN 981-02-4541-6.
https://en.wikipedia.org/wiki/Ring_of_symmetric_functionshttps://en.wikipedia.org/wiki/Monomialhttps://en.wikipedia.org/wiki/Monomial_symmetric_polynomialhttps://en.wikipedia.org/wiki/Power_sum_symmetric_polynomialhttps://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttps://en.wikipedia.org/wiki/Symmetric_functionhttps://en.wikipedia.org/wiki/Fluid_solutionhttps://en.wikipedia.org/wiki/Einstein_tensorhttps://en.wikipedia.org/wiki/Perfect_fluidhttps://en.wikipedia.org/wiki/Exact_solutions_in_general_relativityhttps://en.wikipedia.org/wiki/Exact_solutions_in_general_relativityhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2004999https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/981-02-4541-6


Tignol, Jean-Pierre (2001). Galois theory of algebraic equations. Singapore: World Scientific. ISBN 978-981-02-4541-2.

Bergeron, F.; Labelle, G.; and Leroux, P. (1998). Combinatorial species and tree-like structures. Cambridge:Cambridge University Press. ISBN 978-0-521-57323-8.

Cameron, Peter J. (1999). Permutation Groups. Cambridge: Cambridge University Press. ISBN 978-0-521-65378-7.

Cox, David; Little, John, and O'Shea, Donal (1992). Ideals, Varieties, and Algorithms. New York: Springer-Verlag. ISBN 978-0-387-97847-5.

Eppstein, D.; Goodrich, M. T. (2007). Space-efficient straggler identification in round-trip data streams viaNewtons identities and invertible Bloom filters. Algorithms and Data Structures, 10th International Workshop,WADS 2007. Springer-Verlag, Lecture Notes in Computer Science 4619. pp. 637648. arXiv:0704.3313

Littlewood, D. E. (1950). The theory of group characters and matrix representations of groups. Oxford: OxfordUniversity Press. viii+310. ISBN 0-8218-4067-3.

Macdonald, I. G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs.Oxford: The Clarendon Press, Oxford University Press. viii+180. ISBN 0-19-853530-9. MR 84g:05003.

Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Sec-ond ed.). New York: Oxford Science Publications. The Clarendon Press, Oxford University Press. p. x+475.ISBN 0-19-853489-2. MR 96h:05207.

Mead, D.G. (1992-10). Newtons Identities. The American Mathematical Monthly (Mathematical Associ-ation of America) 99 (8): 749751. doi:10.2307/2324242. JSTOR 2324242. Check date values in: |date=(help)

Stanley, Richard P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press. ISBN 0-521-56069-1. (hardback). ISBN 0-521-78987-7 (paperback).

Sturmfels, Bernd (1992). Algorithms in Invariant Theory. New York: Springer-Verlag. ISBN 978-0-387-82445-1.

Tucker, Alan (1980). Applied Combinatorics (5/e ed.). New York: Wiley. ISBN 978-0-471-73507-6.

1.6 External links NewtonGirard formulas on MathWorld

A Matrix Proof of Newtons Identities in Mathematics Magazine

Application on the number of real roots
https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-981-02-4541-2https://en.wikipedia.org/wiki/Special:BookSources/978-981-02-4541-2https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-57323-8https://en.wikipedia.org/wiki/Peter_Cameron_(mathematician)https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-65378-7https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-65378-7https://en.wikipedia.org/wiki/David_A._Coxhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97847-5https://en.wikipedia.org/wiki/David_Eppsteinhttps://en.wikipedia.org/wiki/Michael_T._Goodrichhttps://en.wikipedia.org/wiki/Workshop_on_Algorithms_and_Data_Structureshttps://en.wikipedia.org/wiki/Workshop_on_Algorithms_and_Data_Structureshttps://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/0704.3313https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-8218-4067-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-19-853530-9https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=84g%253A05003https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-19-853489-2https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=96h%253A05207https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2324242https://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/2324242https://en.wikipedia.org/wiki/Help:CS1_errors#bad_datehttps://en.wikipedia.org/wiki/Richard_P._Stanleyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-56069-1https://en.wikipedia.org/wiki/Special:BookSources/0-521-56069-1https://en.wikipedia.org/wiki/Special:BookSources/0521789877https://en.wikipedia.org/wiki/Bernd_Sturmfelshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-82445-1https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-82445-1https://en.wikipedia.org/wiki/Alan_Tuckerhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-471-73507-6http://mathworld.wolfram.com/Newton-GirardFormulas.htmlhttp://links.jstor.org/sici?sici=0025-570X(200010)73%253A4%253C313%253AAMPONI%253E2.0.CO%253B2-0http://stellar.mit.edu/S/course/6/sp10/6.256/courseMaterial/topics/topic2/lectureNotes/lecture-05/lecture-05.pdf

Chapter 2

Nullform

In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of thegroup vanish. Nullforms were introduced by Hilbert (1893). (Dieudonn & Carrell 1970, 1971, p.57).

2.1 References Dieudonn, Jean A.; Carrell, James B. (1970), Invariant theory, old and new, Advances in Mathematics 4:180, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525

Dieudonn, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

Hilbert, David (1893), Ueber die vollen Invariantensysteme,Mathematische Annalen (Springer Berlin / Hei-delberg) 42: 313373, doi:10.1007/BF01444162, ISSN 0025-5831

11
https://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Group_actionhttps://en.wikipedia.org/wiki/Linear_maphttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Invariant_theoryhttps://en.wikipedia.org/wiki/David_Hilberthttps://en.wikipedia.org/wiki/Nullform#CITEREFHilbert1893https://en.wikipedia.org/wiki/Nullform#CITEREFDieudonn.C3.A9Carrell1970https://en.wikipedia.org/wiki/Nullform#CITEREFDieudonn.C3.A9Carrell1971https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1016%252F0001-8708%252870%252990015-0https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0001-8708https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0255525https://en.wikipedia.org/wiki/Academic_Presshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1016%252F0001-8708%252870%252990015-0https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-12-215540-6https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0279102https://en.wikipedia.org/wiki/David_Hilberthttps://en.wikipedia.org/wiki/Mathematische_Annalenhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF01444162https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0025-5831

Chapter 3

Osculant

In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurfacethat vanishes if the hypersurface touches itself, or an invariant of several hypersurfaces that osculate, meaning thatthey have a common point where they meet to unusually high order.

3.1 References Salmon, George (1885) [1859], Lessons introductory to the modern higher algebra (4th ed.), Dublin, Hodges,Figgis, and Co., ISBN 978-0-8284-0150-0

12
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Chapter 4

Perpetuant

In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or infinite degree.More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary formstabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space arecalled perpetuants. Perpetuants were introduced and named by Sylvester (1882, p.105). MacMahon (1884, 1885,1894) and Stroh (1888) classified the perpetuants. Elliott (1907) describes the early history of perpetuants and givesan annotated bibliography. There are very few papers after about 1910 discussing perpetuants; (Littlewood 1944) isone of the few exceptions.MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weightw is the coefficient of xw of

x2n11

(1 x2)(1 x3) (1 xn)

For n=1 there is just one perpetuant, of weight 0, and for n=2 the number is given by the coefficient of xw of x2/(1-x2).

4.1 References

Cayley, Arthur (1884), A Memoir on Seminvariants, American Journal of Mathematics (The Johns HopkinsUniversity Press) 7 (1): 125, doi:10.2307/2369456, ISSN 0002-9327

Elliott, Edwin Bailey (1895), An introduction to the algebra of quantics, Oxford, Clarendon Press, Reprintedby Chelsea Scientific Books 1964

Elliott, Edwin Bailey (1907), On Perpetuants and Contra-Perpetuants, Proc. London Math. Soc. 4 (1):228246, doi:10.1112/plms/s2-4.1.228

Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge University Press

Littlewood, D. E. (1944), Invariant theory, tensors and group characters, Philosophical Transactions of theRoyal Society of London. Series A.Mathematical and Physical Sciences 239: 305365, doi:10.1098/rsta.1944.0001,ISSN 0080-4614, MR 0010594

MacMahon, P. A. (1884), On Perpetuants, American Journal of Mathematics (The Johns Hopkins UniversityPress) 7 (1): 2646, doi:10.2307/2369457, ISSN 0002-9327

MacMahon, P. A. (1885), A Second Paper on Perpetuants, American Journal of Mathematics (The JohnsHopkins University Press) 7 (3): 259263, doi:10.2307/2369271, ISSN 0002-9327

MacMahon, P. A. (1894), The Perpetuant Invariants of Binary Quantics, Proc. London Math. Soc. 26 (1):262284, doi:10.1112/plms/s1-26.1.262

13
https://en.wikipedia.org/wiki/Invariant_theoryhttps://en.wikipedia.org/wiki/Perpetuant#CITEREFSylvester1882https://en.wikipedia.org/wiki/Perpetuant#CITEREFMacMahon1884https://en.wikipedia.org/wiki/Perpetuant#CITEREFMacMahon1885https://en.wikipedia.org/wiki/Perpetuant#CITEREFMacMahon1894https://en.wikipedia.org/wiki/Perpetuant#CITEREFStroh1888https://en.wikipedia.org/wiki/Perpetuant#CITEREFElliott1907https://en.wikipedia.org/wiki/Perpetuant#CITEREFLittlewood1944https://en.wikipedia.org/wiki/Arthur_Cayleyhttp://www.jstor.org/stable/2369456https://en.wikipedia.org/wiki/American_Journal_of_Mathematicshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2369456https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9327http://archive.org/details/introductiontoal00elliialahttp://books.google.com/books?id=a7xLAAAAYAAJ&pg=PA228https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1112%252Fplms%252Fs2-4.1.228https://en.wikipedia.org/wiki/Alfred_Younghttp://archive.org/details/algebraofinvaria00gracialahttps://en.wikipedia.org/wiki/Cambridge_University_Presshttp://www.jstor.org/stable/91389https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1098%252Frsta.1944.0001https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0080-4614https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0010594http://www.jstor.org/stable/2369457https://en.wikipedia.org/wiki/American_Journal_of_Mathematicshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2369457https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9327http://www.jstor.org/stable/2369271https://en.wikipedia.org/wiki/American_Journal_of_Mathematicshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2369271https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9327https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1112%252Fplms%252Fs1-26.1.262

14 CHAPTER 4. PERPETUANT

Stroh, E. (1888), Ueber eine fundamentale Eigenschaft des Ueberschiebungs-processes und deren Verw-erthung in der Theorie der binren Formen, Mathematische Annalen (Springer Berlin / Heidelberg) 33: 61107, doi:10.1007/bf01444111, ISSN 0025-5831

Sylvester, James Joseph (1882), On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Or-der,American Journal ofMathematics (The JohnsHopkinsUniversity Press) 5 (1): 79136, doi:10.2307/2369536,ISSN 0002-9327
http://dx.doi.org/10.1007/BF01444111http://dx.doi.org/10.1007/BF01444111https://en.wikipedia.org/wiki/Mathematische_Annalenhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252Fbf01444111https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0025-5831http://www.jstor.org/stable/2369536http://www.jstor.org/stable/2369536https://en.wikipedia.org/wiki/American_Journal_of_Mathematicshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2369536https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9327

Chapter 5

Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set ofpolynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, oftena field. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the constructionof splitting fields, and to the understanding of a linear operator. Many important conjectures involving polynomialrings, such as Serres problem, have influenced the study of other rings, and have influenced even the definition ofother rings, such as group rings and rings of formal power series.A closely related notion is that of the ring of polynomial functions on a vector space.

5.1 The polynomial ring K[X]

5.1.1 Definition

The polynomial ring, K[X], in X over a field K is defined[1] as the set of expressions, called polynomials in X, ofthe form

p = p0 + p1X + p2X2 + + pm1Xm1 + pmXm,

where p0, p1,, p , the coefficients of p, are elements of K, and X, X 2, are formal symbols (the powers of X"). Byconvention, X 0 = 1, X 1 = X, and the product of the powers of X is defined by the familiar formula

Xk X l = Xk+l,

where k and l are any natural numbers. The symbol X is called an indeterminate[2] or variable.[3]

Two polynomials are defined to be equal if and only if the corresponding coefficients for each power of X are equal,however terms with zero coefficient, 0X k, may be added or omitted.This terminology is suggested by real or complex polynomial functions. However, in general, X and its powers, X k,are treated as formal symbols, not as elements of the field K or functions over it. One can think of the ring K[X] asarising from K by adding one new element X that is external to K and requiring that X commute with all elements ofK.Polynomials in X are added and multiplied according to the ordinary rules for manipulating algebraic expressions,creating the structure of a ring. Specifically, if

p = p0 + p1X + p2X2 + + pmXm,

and

15
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16 CHAPTER 5. POLYNOMIAL RING

q = q0 + q1X + q2X2 + + qnXn,

then

p+ q = r0 + r1X + r2X2 + + rkXk,

and

pq = s0 + s1X + s2X2 + + slX l,

where

ri = pi + qi

and

si = p0qi + p1qi1 + + piq0.

If necessary, the polynomials p and q are extended by adding dummy terms with zero coefficients, so that theexpressions for ri and si are always defined.A more rigorous, but less intuitive treatment[4] defines a polynomial as an infinite tuple, or ordered sequence of ele-ments of K, (p0, p1, p2, ) having the property that only a finite number of the elements are nonzero, or equivalently,a sequence for which there is some m so that pn = 0 for n>m. In this case, the expression

p0 + p1X + p2X2 + + pmXm

is considered an alternate notation for the sequence (p0, p1, p2, pm, 0, 0, ).More generally, the field K can be replaced by any commutative ring R when taking the same construction as above,giving rise to the polynomial ring over R, which is denoted R[X].

5.1.2 Degree of a polynomial

The degree of a polynomial p, written deg(p) is the largest k such that the coefficient of X k is not zero.[5] In this casethe coefficient pk is called the leading coefficient.[6] In the special case of zero polynomial, all of whose coefficientsare zero, the degree has been variously left undefined,[7] defined to be 1,[8] or defined to be a special symbol .[9]

If K is a field, or more generally an integral domain, then from the definition of multiplication,[10]

deg(pq) = deg(p) + deg(q).

It follows immediately that if K is an integral domain then so is K[X].[11]

5.1.3 Properties of K[X]

Factorization in K[X]

The next property of the polynomial ring is much deeper. Already Euclid noted that every positive integer can beuniquely factored into a product of primes this statement is now called the fundamental theorem of arithmetic.The proof is based on Euclids algorithm for finding the greatest common divisor of natural numbers. At each step
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5.1. THE POLYNOMIAL RING K[X] 17

of this algorithm, a pair (a, b), a > b, of natural numbers is replaced by a new pair (b, r), where r is the remainderfrom the division of a by b, and the new numbers are smaller. Gauss remarked that the procedure of division withthe remainder can also be defined for polynomials: given two polynomials p and q, where q 0, one can write

p = uq + r,

where the quotient u and the remainder r are polynomials, the degree of r is less than the degree of q, and a de-composition with these properties is unique. The quotient and the remainder are found using the polynomial longdivision. The degree of the polynomial now plays a role similar to the absolute value of an integer: it is strictly lessin the remainder r than it is in q, and when repeating this step such decrease cannot go on indefinitely. Thereforeeventually some division will be exact, at which point the last non-zero remainder is the greatest common divisorof the initial two polynomials. Using the existence of greatest common divisors, Gauss was able to simultaneouslyrigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. In factthere exist other commutative rings than Z and K[X] that similarly admit an analogue of the Euclidean algorithm;all such rings are called Euclidean rings. Rings for which there exists unique (in an appropriate sense) factoriza-tion of nonzero elements into irreducible factors are called unique factorization domains or factorial rings; the givenconstruction shows that all Euclidean rings, and in particular Z and K[X], are unique factorization domains.Another corollary of the polynomial division with the remainder is the fact that every proper ideal I of K[X] isprincipal, i.e. I consists of the multiples of a single polynomial f. Thus the polynomial ring K[X] is a principalideal domain, and for the same reason every Euclidean domain is a principal ideal domain. Also every principalideal domain is a unique-factorization domain. These deductions make essential use of the fact that the polynomialcoefficients lie in a field, namely in the polynomial division step, which requires the leading coefficient of q, whichis only known to be non-zero, to have an inverse. If R is an integral domain that is not a field then R[X] is neither aEuclidean domain nor a principal ideal domain; however it could still be a unique factorization domain (and will beso if and only it R itself is a unique factorization domain, for instance if it is Z or another polynomial ring).

Quotient ring of K[X]

The ring K[X] of polynomials over K is obtained from K by adjoining one element, X. It turns out that any commu-tative ring L containing K and generated as a ring by a single element in addition to K can be described using K[X].In particular, this applies to finite field extensions of K.Suppose that a commutative ring L contains K and there exists an element of L such that the ring L is generated by over K. Thus any element of L is a linear combination of powers of with coefficients in K. Then there is a uniquering homomorphism from K[X] into L which does not affect the elements of K itself (it is the identity map on K)and maps each power of X to the same power of . Its effect on the general polynomial amounts to replacing X with":

(amXm + am1X

m1 + + a1X + a0) = amm + am1m1 + + a1 + a0.

By the assumption, any element of L appears as the right hand side of the last expression for suitable m and elementsa0, , am of K. Therefore, is surjective and L is a homomorphic image of K[X]. More formally, let Ker be thekernel of . It is an ideal of K[X] and by the first isomorphism theorem for rings, L is isomorphic to the quotientof the polynomial ring K[X] by the ideal Ker . Since the polynomial ring is a principal ideal domain, this ideal isprincipal: there exists a polynomial pK[X] such that

L K[X]/(p).

A particularly important application is to the case when the larger ring L is a field. Then the polynomial p must beirreducible. Conversely, the primitive element theorem states that any finite separable field extension L/K can begenerated by a single element L and the preceding theory then gives a concrete description of the field L as thequotient of the polynomial ring K[X] by a principal ideal generated by an irreducible polynomial p. As an illustration,the field C of complex numbers is an extension of the field R of real numbers generated by a single element i suchthat i2 + 1 = 0. Accordingly, the polynomial X2 + 1 is irreducible over R and
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C R[X]/(X2 + 1).

More generally, given a (not necessarily commutative) ring A containing K and an element a of A that commuteswith all elements of K, there is a unique ring homomorphism from the polynomial ring K[X] to A that maps X to a:

: K[X] A, (X) = a.

This homomorphism is given by the same formula as before, but it is not surjective in general. The existence anduniqueness of such a homomorphism expresses a certain universal property of the ring of polynomials in one variableand explains ubiquity of polynomial rings in various questions and constructions of ring theory and commutativealgebra.

5.1.4 Modules

The structure theorem for finitely generated modules over a principal ideal domain applies over K[X]. This meansthat every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitelymany modules of the formK[X]/P k , where P is an irreducible polynomial over K and k a positive integer.

5.2 Polynomial evaluation

Let K be a field or, more generally, a commutative ring, and R a ring containing K. For any polynomial P in K[X]and any element a in R, the substitution of X by a in P defines an element of R, which is denoted P(a). This elementis obtained by, after the substitution, carrying on, in R, the operations indicated by the expression of the polynomial.This computation is called the evaluation of P at a. For example, if we have

P = X2 1,

we have

P (3) = 32 1 = 8,

P (X2 + 1) = (X2 + 1)2 1 = X4 + 2X2

(in the first example R = K, and in the second one R = K[X]). Substituting X by itself results in

P = P (X),

explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent.For every a in R, the map P 7 P (a) defines a ring homomorphism from K[X] into R.The polynomial function defined by a polynomial P is the function from K into K that is defined by x 7 P (x). If Kis an infinite field, two different polynomials define different polynomial functions, but this property is false for finitefields. For example, if K is a field with q elements, then the polynomials 0 and Xq-X both define the zero function.

5.3 The polynomial ring in several variables

5.3.1 Polynomials

A polynomial in n variables X1, , Xn with coefficients in a field K is defined analogously to a polynomial in onevariable, but the notation is more cumbersome. For any multi-index = (1, , n), where each i is a non-negativeinteger, let
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5.3. THE POLYNOMIAL RING IN SEVERAL VARIABLES 19

X =

ni=1

Xii = X11 . . . X

nn .

The product X is called themonomial of multidegree . A polynomial is a finite linear combination of monomialswith coefficients in K

p =

pX,

where p = p1,...,n K, and only finitely many coefficients p are different from 0. The degree of a monomialX, frequently denoted ||, is defined as

|| =n

i=1

i,

and the degree of a polynomial p is the largest degree of a monomial occurring with non-zero coefficient in theexpansion of p.

5.3.2 The polynomial ring

Polynomials in n variables with coefficients in K form a commutative ring denoted K[X1,, Xn], or sometimesK[X], where X is a symbol representing the full set of variables, X = (X1,, Xn), and called the polynomial ring inn variables. The polynomial ring in n variables can be obtained by repeated application of K[X] (the order by whichis irrelevant). For example, K[X1, X2] is isomorphic to K[X1][X2]. This ring plays fundamental role in algebraicgeometry. Many results in commutative and homological algebra originated in the study of its ideals and modulesover this ring.A polynomial ring with coefficients in Z is the free commutative ring over its set of variables.

5.3.3 Hilberts Nullstellensatz

Main article: Hilberts Nullstellensatz

A group of fundamental results concerning the relation between ideals of the polynomial ring K[X1,, Xn] andalgebraic subsets of Kn originating with David Hilbert is known under the name Nullstellensatz (literally: zero-locus theorem).

(Weak form, algebraically closed field of coefficients). Let K be an algebraically closed field. Then everymaximal ideal m of K[X1,, Xn] has the form

m = (X1 a1, . . . , Xn an), a = (a1, . . . , an) Kn.

(Weak form, any field of coefficients). Let k be a field, K be an algebraically closed field extension of k, and Ibe an ideal in the polynomial ring k[X1,, Xn]. Then I contains 1 if and only if the polynomials in I do nothave any common zero in Kn.

(Strong form). Let k be a field, K be an algebraically closed field extension of k, I be an ideal in the polynomialring k[X1,, Xn],and V(I) be the algebraic subset of Kn defined by I. Suppose that f is a polynomial whichvanishes at all points of V(I). Then some power of f belongs to the ideal I:
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fm I, some form N.

Using the notion of the radical of an ideal, the conclusion says that f belongs to the radical of I. As acorollary of this form of Nullstellensatz, there is a bijective correspondence between the radical idealsof K[X1,, Xn] for an algebraically closed field K and the algebraic subsets of the n-dimensional affinespace Kn. It arises from the map

I 7 V (I), I K[X1, . . . , Xn], V (I) Kn.

The prime ideals of the polynomial ring correspond to irreducible subvarieties of Kn.

5.4 Properties of the ring extension R R[X]

One of the basic techniques in commutative algebra is to relate properties of a ring with properties of its subrings. Thenotation R S indicates that a ring R is a subring of a ring S. In this case S is called an overring of R and one speaksof a ring extension. This works particularly well for polynomial rings and allows one to establish many importantproperties of the ring of polynomials in several variables over a field, K[X1,, Xn], by induction in n.

5.4.1 Summary of the results

In the following properties, R is a commutative ring and S = R[X1,, Xn] is the ring of polynomials in n variablesover R. The ring extension R S can be built from R in n steps, by successively adjoining X1,, Xn. Thus to establisheach of the properties below, it is sufficient to consider the case n = 1.

If R is an integral domain then the same holds for S.

If R is a unique factorization domain then the same holds for S. The proof is based on the Gauss lemma.

Hilberts basis theorem: If R is a Noetherian ring, then the same holds for S.

Suppose that R is a Noetherian ring of finite global dimension. Then

gl dimR[X1, . . . , Xn] = gl dimR+ n.

An analogous result holds for Krull dimension.

5.5 Generalizations

Polynomial rings have been generalized in a great many ways, including polynomial rings with generalized exponents,power series rings, noncommutative polynomial rings, and skew-polynomial rings.

5.5.1 Infinitely many variables

The possibility to allow an infinite set of indeterminates is not really a generalization, as the ordinary notion of poly-nomial ring allows for it. It is then still true that each monomial involves only a finite number of indeterminates(so that its degree remains finite), and that each polynomial is a linear combination of monomials, which by defini-tion involves only finitely many of them. This explains why such polynomial rings are relatively seldom considered:
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5.5. GENERALIZATIONS 21

each individual polynomial involves only finitely many indeterminates, and even any finite computation involvingpolynomials remains inside some subring of polynomials in finitely many indeterminates.In the case of infinitely many indeterminates, one can consider a ring strictly larger than the polynomial ring butsmaller than the power series ring, by taking the subring of the latter formed by power series whose monomials havea bounded degree. Its elements still have a finite degree and are therefore are somewhat like polynomials, but it ispossible for instance to take the sum of all indeterminates, which is not a polynomial. A ring of this kind plays a rolein constructing the ring of symmetric functions.

5.5.2 Generalized exponents

Main article: Monoid ring

A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas foraddition and multiplication make sense as long as one can add exponents: XiXj = Xi+j . A set for which additionmakes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R whichare nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of Nwith coefficients in R. The addition is defined component-wise, so that if c = a+b, then cn = an + bn for every n in N.The multiplication is defined as the Cauchy product, so that if c = ab, then for each n in N, cn is the sum of all aibjwhere i, j range over all pairs of elements of N which sum to n.When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:

nN

anXn

and then the formulas for addition and multiplication are the familiar:

(nN

anXn

)+

(nN

bnXn

)=nN

(an + bn)Xn

and

(nN

anXn

)

(nN

bnXn

)=nN

i+j=n

aibj

Xnwhere the latter sum is taken over all i, j in N that sum to n.Some authors such as (Lang 2002, II,3) go so far as to take this monoid definition as the starting point, and regularsingle variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials inseveral variables simply take N to be the direct product of several copies of the monoid of non-negative integers.Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negativerational numbers, (Osbourne 2000, 4.4).

5.5.3 Power series

Main article: Formal power series

Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. Thisrequires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy productare finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums.For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series isdefined as the set of functions from N to a ring R with addition component-wise, and multiplication given by theCauchy product. The ring of power series can be seen as the completion of the polynomial ring.
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5.5.4 Noncommutative polynomial rings

Main article: Free algebra

For polynomial rings of more than one variable, the products XY and YX are simply defined to be equal. A moregeneral notion of polynomial ring is obtained when the distinction between these two formal products is maintained.Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the monoid ring R[N],where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of nsymbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongstthemselves, but the coefficients and variables commute with each other.Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebraof rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the freeassociative, unital R-algebra on n generators, which is noncommutative when n > 1.

5.5.5 Differential and skew-polynomial rings

Main article: Ore extension

Other generalizations of polynomials are differential and skew-polynomial rings.A differential polynomial ring is a ring of differential operators formed from a ring R and a derivation of R into R.This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate onR by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation(ab) = a(b) + (a)b may be rewritten

X a = a X + (a).

This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R,which make them a non-commutative ring.The standard example, called a Weyl algebra, takes R to be a (usual) polynomial ring k[Y], and to be the standardpolynomial derivative Y . Taking a =Y in the above relation, one gets the canonical commutation relation,XY Y X= 1. Extending this relation by associativity and distributivity allows to construct explicitly the Weyl algebra.(Lam2001, 1,ex1.9).The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending themultiplication from the relationXr = f(r)X to produce an associative multiplication that distributes over the standardaddition. More generally, given a homomorphismF from themonoidN of the positive integers into the endomorphismring of R, the formula Xnr = F(n)(r)Xn allows to construct a skew-polynomial ring.(Lam 2001, 1,ex 1.11) Skewpolynomial rings are closely related to crossed product algebras.

5.6 See also Additive polynomial

Laurent polynomial

5.7 References[1] Following Herstein p. 153

[2] Herstein, Hall p. 73

[3] Lang p. 97

[4] Following Hall p.72-73
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5.7. REFERENCES 23

[5] Herstein p. 154

[6] Lang p.100

[7] Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), Calculus Single Variable, John Wiley & Sons, p. 31, ISBN9780470647707.

[8] Sendra, J. Rafael; Winkler, Franz; Prez-Diaz, Sonia (2007), Rational Algebraic Curves: A Computer Algebra Approach,Algorithms and Computation in Mathematics 22, Springer, p. 250, ISBN 9783540737247.

[9] Eves, Howard Whitley (1980), Elementary Matrix Theory, Dover, p. 183, ISBN 9780486150277.

[10] Herstein p.155, 162

[11] Herstein p.162

Hall, F. M. (1969). Section 3.6. An Introduction to Abstract Algebra 2. Cambridge University Press. ISBN0521084849.

Herstein, I. N. (1975). Section 3.9. Topics in Algebra. Wiley. ISBN 0471010901.

Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN978-0-387-95325-0

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics 196, Berlin, New York:Springer-Verlag, ISBN 978-0-387-98934-1, MR 1757274
http://books.google.com/books?id=U2uv84cpJHQC&pg=RA1-PA31https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/9780470647707http://books.google.com/books?id=puWxs7KG2D0C&pg=PA250https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/9783540737247https://en.wikipedia.org/wiki/Howard_Eveshttp://books.google.com/books?id=ayVxeUNbZRAC&pg=PA183https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/9780486150277https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0521084849https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0471010901https://en.wikipedia.org/wiki/Tsit_Yuen_Lamhttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95325-0https://en.wikipedia.org/wiki/Serge_Langhttps://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematicshttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95385-4https://en.wikipedia.org/wiki/Mathematical_Reviewshttp://www.ams.org/mathscinet-getitem?mr=1878556https://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98934-1https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1757274

Chapter 6

Quantum invariant

In the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jonespolynomial of surgery presentations of the knot complement.[1] [2] [3]

6.1 List of invariants Finite type invariant

Kontsevich invariant

Kashaevs invariant

WittenReshetikhinTuraev invariant (ChernSimons)

Invariant differential operator[4]

RozanskyWitten invariant

Vassiliev knot invariant

Dehn invariant

LMO invariant [5]

TuraevViro invariant

DijkgraafWitten invariant [6]

ReshetikhinTuraev invariant

Tau-invariant

I-Invariant

Klein J-invariant

Quantum isotopy invariant [7]

ErmakovLewis invariant

Hermitian invariant

GoussarovHabiro theory of finite-type invariant

Linear quantum invariant (orthogonal function invariant)

MurakamiOhtsuki TQFT

Generalized Casson invariant

24
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6.2. SEE ALSO 25

Casson-Walker invariant

KhovanovRozansky invariant

HOMFLY polynomial

K-theory invariants

AtiyahPatodiSinger eta invariant

Link invariant [8]

Casson invariant

SeibergWitten invariant

GromovWitten invariant

Arf invariant

Hopf invariant

6.2 See also Invariant theory

Framed knot

ChernSimons theory

Algebraic geometry

Seifert surface

Geometric invariant theory

6.3 References[1] Reshetikhin, N. & Turaev, V. (1991). Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.

103 (1): 547. doi:10.1007/BF01239527. Retrieved 4 December 2012.

[2] Kontsevich, Maxim (1993). Vassilievs knot invariants. Adv. Soviet Math. 16: 137.

[3] Watanabe, Tadayuki (2007). Knotted trivalent graphs and construction of the LMO invariant from triangulations. OsakaJ. Math. 44 (2): 351. Retrieved 4 December 2012.

[4] [math/0406194] Invariant differential operators for quantum symmetric spaces, II

[5] [math/0009222v1] Topological quantum field theory and hyperk\"ahler geometry

[6] http://hal.archives-ouvertes.fr/docs/00/09/02/99/PDF/equality_arxiv_1.pdf

[7] http://knot.kaist.ac.kr/7thkgtf/Lawton1.pdf

[8] Invariants of 3-manifolds via link polynomials and quantum groups - Springer

6.4 Further reading Freedman, Michael H. (1990). Topology of 4-manifolds. Princeton, N.J: Princeton University Press. ISBN0691085773.

Ohtsuki, Tomotada (December 2001). Quantum Invariants. World Scientific Publishing Company. ISBN9789810246754.
https://en.wikipedia.org/wiki/Casson-Walker_invarianthttps://en.wikipedia.org/wiki/HOMFLY_polynomialhttps://en.wikipedia.org/wiki/Atiyah%E2%80%93Patodi%E2%80%93Singer_eta_invarianthttps://en.wikipedia.org/wiki/Knot_invarianthttps://en.wikipedia.org/wiki/Casson_invarianthttps://en.wikipedia.org/wiki/Seiberg%E2%80%93Witten_invarianthttps://en.wikipedia.org/wiki/Gromov%E2%80%93Witten_invarianthttps://en.wikipedia.org/wiki/Arf_invariant_(knot)https://en.wikipedia.org/wiki/Hopf_invarianthttps://en.wikipedia.org/wiki/Invariant_theoryhttps://en.wikipedia.org/wiki/Framed_knothttps://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theoryhttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Seifert_surfacehttps://en.wikipedia.org/wiki/Geometric_invariant_theoryhttp://link.springer.com/content/pdf/10.1007%252FBF01239527https://en.wikipedia.org/wiki/Inventiones_Mathematicaehttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF01239527http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ojm/1183667985http://arxiv.org/abs/math.QA/0406194http://arxiv.org/abs/math/0009222v1http://hal.archives-ouvertes.fr/docs/00/09/02/99/PDF/equality_arxiv_1.pdfhttp://knot.kaist.ac.kr/7thkgtf/Lawton1.pdfhttp://www.springerlink.com/content/u416971m947560r7/fulltext.pdfhttp://openlibrary.org/books/OL2220094M/Topology_of_4-manifoldshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0691085773http://openlibrary.org/books/OL9195378M/Quantum_Invariantshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/9789810246754

26 CHAPTER 6. QUANTUM INVARIANT

6.5 External links Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev
http://books.google.com.mx/books?id=yQRDNCJ0iOUC

Chapter 7

Quaternary cubic

In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros forma cubic surface in 3-dimensional projective space.

7.1 Invariants

Salmon (1860) and Clebsch (1861, 1861b) studied the ring of invariants of a quaternary cubic, which is a ringgenerated by invariants of degrees 8, 16, 24, 32, 40, 100. The generators of degrees 8, 16, 24, 32, 40 generatea polynomial ring. The generator of degree 100 is a skew invariant, whose square is a polynomial in the othergenerators given explicitly by Salmon. Salmon also gave an explicit formula for the discriminant as a polynomial inthe generators, though Edge (1980) pointed out that the formula has a widely-copied misprint in it.

7.2 Sylvester pentahedron

A generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cuberoots of unity. This was conjectured by Sylvester in 1851, and proven 10 years later by Clebsch. The union of the 5planes where these 5 linear forms vanish is called the Sylvester pentahedron.

7.3 See also

Ternary cubic

Ternary quartic

Invariants of a binary form

7.4 References

Clebsch, A. (1861), Zur Theorie der algebraischer Flchen, Journal fr die reine und angewandteMathematik58: 93108, ISSN 0075-4102

Clebsch, A. (1861), Ueber eine Transformation der homogenen Funktionen dritter Ordnung mit vier Vern-derlichen, Journal fr die reine und angewandte Mathematik 58: 109126, doi:10.1515/crll.1861.58.109,ISSN 0075-4102

Edge, W. L. (1980), The Discriminant of a Cubic Surface, Proceedings of the Royal Irish Academy (RoyalIrish Academy) 80A (1): 7578, ISSN 0035-8975

27
https://en.wikipedia.org/wiki/Cubic_surfacehttps://en.wikipedia.org/wiki/Quaternary_cubic#CITEREFSalmon1860https://en.wikipedia.org/wiki/Quaternary_cubic#CITEREFClebsch1861https://en.wikipedia.org/wiki/Quaternary_cubic#CITEREFClebsch1861bhttps://en.wikipedia.org/wiki/Quaternary_cubic#CITEREFEdge1980https://en.wikipedia.org/wiki/Ternary_cubichttps://en.wikipedia.org/wiki/Ternary_quartichttps://en.wikipedia.org/wiki/Invariants_of_a_binary_formhttp://resolver.sub.uni-goettingen.de/purl?GDZPPN002151049https://en.wikipedia.org/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematikhttps://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0075-4102http://resolver.sub.uni-goettingen.de/purl?GDZPPN002151057http://resolver.sub.uni-goettingen.de/purl?GDZPPN002151057https://en.wikipedia.org/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematikhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1515%252Fcrll.1861.58.109https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0075-4102http://www.jstor.org/stable/20489083https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0035-8975

28 CHAPTER 7. QUATERNARY CUBIC

Salmon, George (1860), On Quaternary Cubics, Philosophical Transactions of the Royal Society (The RoyalSociety) 150: 229239, doi:10.1098/rstl.1860.0015, ISSN 0080-4614

Schmitt, Alexander (1997), Quaternary cubic forms and projective algebraic threefolds, L'EnseignementMathmatique. Revue Internationale. IIe Srie 43 (3): 253270, ISSN 0013-8584, MR 1489885
http://www.jstor.org/stable/108770https://en.wikipedia.org/wiki/Philosophical_Transactions_of_the_Royal_Societyhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1098%252Frstl.1860.0015https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0080-4614http://retro.seals.ch/digbib/view?rid=ensmat-001:1997:43::125&id=hitlisthttps://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0013-8584https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1489885

Chapter 8

Quippian

In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Cayley (1857) anddiscussed by Dolgachev (2012, p.157). In the same paper Cayley also introduced another similar invariant that hecalled the pippian, now called the Cayleyan.

8.1 See also Glossary of classical algebraic geometry

8.2 References Cayley, Arthur (1857), A Memoir on Curves of the Third Order, Philosophical Transactions of the RoyalSociety of London (The Royal Society) 147: 415446, doi:10.1098/rstl.1857.0021, ISSN 0080-4614

Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view, Cambridge University Press, ISBN978-1-107-01765-8

29
https://en.wikipedia.org/wiki/Quippian#CITEREFCayley1857https://en.wikipedia.org/wiki/Quippian#CITEREFDolgachev2012https://en.wikipedia.org/wiki/Cayleyanhttps://en.wikipedia.org/wiki/Glossary_of_classical_algebraic_geometryhttps://en.wikipedia.org/wiki/Arthur_Cayleyhttp://www.jstor.org/stable/108626https://en.wikipedia.org/wiki/Philosophical_Transactions_of_the_Royal_Society_of_Londonhttps://en.wikipedia.org/wiki/Philosophical_Transactions_of_the_Royal_Society_of_Londonhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1098%252Frstl.1857.0021https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0080-4614https://en.wikipedia.org/wiki/Igor_Dolgachevhttp://www.math.lsa.umich.edu/~idolga/CAG.pdfhttps://en.wikipedia.org/wiki/Cambridge_University_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-107-01765-8

30 CHAPTER 8. QUIPPIAN

8.3 Text and image sources, contributors, and licenses

8.3.1 Text Newtons identities Source: https://en.wikipedia.org/wiki/Newton{}s_identities?oldid=667597001Contributors: Michael Hardy, Charles

Matthews, Chuunen Baka, Giftlite, Icairns, Rich Farmbrough, Qutezuce, Paul August, Zaslav, Gauge, Reinyday, LutzL, Tony Sidaway, Kb-dank71, Rjwilmsi, Ligulem, Mathbot, JYOuyang, Hillman, KSmrq, Edinborgarstefan, That Guy, From That Show!, SmackBot, Mhym,Druseltal2005, DA3N, Makyen, Simon12, A. Pichler, Thijs!bot, Wikid77, Dogaroon, Headbomb, AntiVandalBot, T0, Shlomi Hillel,Salgueiro~enwiki, David Eppstein, Plasticup, Quantling, Jszigeti, Mulanhua, AlleborgoBot, JerroldPease-Atlanta, Prohlep, Mild Bill Hic-cup, Razorflame, Marc van Leeuwen, Addbot, DOI bot, Lightbot, Andreasmperu, AnomieBOT, Citation bot, Citation bot 1, BertSeghers,EmausBot, John of Reading, ZroBot, Tttfffkkk, KlappCK, BG19bot, Boriaj, Paolo Lipparini, Brad7777, Rongator, Teddyktchan, SoSivrand Anonymous: 32

Nullform Source: https://en.wikipedia.org/wiki/Nullform?oldid=627020191 Contributors: Rjwilmsi, R.e.b., TexasAndroid, David Epp-stein and Trappist the monk

Osculant Source: https://en.wikipedia.org/wiki/Osculant?oldid=548073772 Contributors: R.e.b. and David Eppstein Perpetuant Source: https://en.wikipedia.org/wiki/Perpetuant?oldid=627025070 Contributors: Michael Hardy, Rjwilmsi, R.e.b. and

Trappist the monk Polynomial ring Source: https://en.wikipedia.org/wiki/Polynomial_ring?oldid=670385454 Contributors: Michael Hardy, TakuyaMu-

rata, Ahoerstemeier, Charles Matthews, Taxman, Zero0000, MathMartin, Tobias Bergemann, Giftlite, Fropuff, Waltpohl, Jason Quinn,Gauss, Flyhighplato, TheObtuseAngleOfDoom, DonDiego, Rgdboer, Oleg Alexandrov, Imaginatorium, Linas, MFH, Grammarbot,Rjwilmsi, Salix alba, DVdm, Hillman, Mathaxiom~enwiki, Gwaihir, Arthur Rubin, Bo Jacoby, SmackBot, RDBury, Mmernex, Knowled-geOfSelf, MalafayaBot, Silly rabbit, MvH, Waggers, Cydebot, RobHar, David Eppstein, WATARU, LokiClock, Anonymous Dissident,Alephcero~enwiki, Arcfrk, SieBot, Henry Delforn (old), JackSchmidt, Aiden Fisher, Classicalecon, Justin W Smith, , Alexbot, Pix-elBot, Cacadril, Marc van Leeuwen, Algebran, D.M. from Ukraine, Addbot, CountryBot, Hyginsberg, Yobot, Ht686rg90, Cflm001,Calle, AnomieBOT, ArthurBot, DSisyphBot, Point-set topologist, Noamz, DivineAlpha, Kallikanzarid, Trappist the monk, Wikitanvir-Bot, D.Lazard, Solvecolorer, Mark viking and Anonymous: 30

Quantum invariant Source: https://en.wikipedia.org/wiki/Quantum_invariant?oldid=647546489 Contributors: Michael Hardy, Ben-der235, R.e.b., Ilmari Karonen, David Eppstein, Moonriddengirl, HenryDelforn (old), NuclearWarfare, Addbot, EmausBot, Theopolisme,Helpful Pixie Bot, Enyokoyama, Paritto and Hctrmycss

Quaternary cubic Source: https://en.wikipedia.org/wiki/Quaternary_cubic?oldid=635282082 Contributors: Michael Hardy, Rjwilmsi,R.e.b., Sfan00 IMG, Trappist the monk, Anrnusna, Monkbot and Anonymous: 1

Quippian Source: https://en.wikip

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