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Binary Quadratic Forms (BQF)
I Binary quadratic form:
Degree 2 homogeneous polynomial in 2 variables.
f(x, y) = ax2 + bxy + cy2
I Ex. f(x, y) = 3x2 − 2xy + y2.
I Seems easy?
Binary Quadratic Forms (BQF)
I Popular with old-timey Europeans like Fermat, Lagrange,Legendre, Euler, Gauss.
I Classic questions:I Which primes are the sum of two squares?
p = x2 + y2?
I When is a BQF invariant under change of coordinates?
f(x, y) = x2 − xy + y2 = f(x− y, x).
I When are two BQF the same after change of coordinates?
What’s the Big Deal?!
Legendre writing aboutBQF in 1798.
David A. Cox writing aboutBQF in 1989.
What’s the Big Deal?!Conway writing about BQF in 1997!
Introducing: the Topograph
Introducing: the Topograph
Introducing: the Topograph
Introducing: the Topograph
Topographical FactsI No vector ever repeats.I Vectors
[ab
]and
[cd
]are adjacent if and only if
ad− bc = det
[a cb d
]= ±1.
Topographical Facts
Say[ab
]is primitive if gcd(a, b) = 1.
I Every primitive vector shows up in the topograph.
I Define the Top and Bottom operations by
T[ab
]=
[a + bb
]B[ab
]=
[a
a + b
].
I Every primitive vector can be made from[10
]or
[01
]using a
sequence of top and bottom operations.
Topographical Facts
Idea: work backwards![311
]= B3
[32
]= B3T1
[12
]= B3T1B2
[10
].
Can you see why this works? Try more examples!
Topographical Facts
Idea: work backwards![311
]= B3
[32
]= B3T1
[12
]= B3T1B2
[10
].
Can you see why this works? Try more examples!
BQF on the Topograph
f(x, y) = ax2 + bxy + cy2
I f(kx, ky) = a(kx)2 + b(kx)(ky) + c(ky)2 = k2f(x, y)
I f(−x,−y) = f(x, y)
I Enough to know the values of f(x, y) on primitive vectors.
BQF on the Topograph
BQF on the Topograph
The Arithmetic Progression Law
From (Almost) Nothing to Everything!
I We don’t even need to know the vectors!I Coordinate free way to study BQF.
From (Almost) Nothing to Everything!
From (Almost) Nothing to Everything!
From (Almost) Nothing to Everything!
From (Almost) Nothing to Everything!
Formula Found
The Climbing Lemma
The Climbing Lemma
Race to the Bottom
I We can always head downhill!
I What happens at the bottom?
Stuck in the Well
I Climbing Lemma: At most one well.I If f > 0, then it has a well and f is called positive definite.
I To tell if two positive definite forms are equivalent, go meetat the well.
Down by the River
I Climbing Lemma: At most one river!I If f 6= 0 and has a river, then f is called indefinite.
Down by the River
I Climbing Lemma: At most one river!I If f 6= 0 and has a river, then f is called indefinite.
DiscriminationIf a and c are adjacent values of f with common difference hbetween them, then the discriminant of f is
∆ = h2 − 4ac.
I Same value measured anywhere on the topograph!I Ex. ∆ = −8.
River or Well?
The Repeating River
Suppose f has a river.
I Finitely many options for a, c, h.
I Therefore the river must always repeat!
The Repeating River
f(x, y) = x2 + 3xy − 2y2
Lakes
I If f has a lake, then ∆ = h2 is a square.I One lake =⇒ f(x, y) = ax2.
Lakes
If both ± appear, there are two lakes connected by a river.
Lakes
Sometimes the river floods and the lakes merge.
Symmetries
I A symmetry of a BQF f is a linear change of coordinatesthat leaves f invariant,
f(Ax + By,Cx + Dy) = f(x, y).
I Alternatively, it is a transformation of the topograph thatpreserves the f labelling.
SymmetriesI A symmetry of a BQF f is a linear change of coordinates
that leaves f invariant,f(Ax + By,Cx + Dy) = f(x, y).
I Alternatively, it is a transformation of the topograph thatpreserves the f labelling.
SymmetriesSymmetries preserve all special features!
I Wells, rivers, and lakes, are preserved.
Another Perspective
Another Perspective
The mediant of reduced fractions ab and c
d is a + cb + d .
Another Perspective
The mediant of reduced fractions ab and c
d is a + cb + d .
Equivalence
Old Facts, New Light
Two rational numbers ab and c
d are adjacent if and only if theirdifference is an Egyptian fraction 1
m .
ab −
cd =
ad− bcbd = ± 1
bd .
Old Facts, New Light
Every reduced fraction eventually appears.I Define Top and Bottom operations on fractions by
T(ab
)=
a + bb B
(ab
)=
aa + b .
I Notice: T is “add one” and B is “flip, add one, flip.”
T(ab
)= 1 +
ab B
(ab
)=
1a+ba
=1
1 + ba
=1
1 + 1ab
.
I Applying T and B repeatedly is easy.
Tm(ab
)= m +
ab Bm
(ab
)=
1m + 1
ab
.
I (flip, add one, flip)(flip, add one, flip) = (flip, add two, flip)
Continued Fractions
I Recall that [311
]= B3T1B2
[10
].
I Check it out:
B3T1B2( 10
)=
1
3 +1
1 +1
2 +1∞
=1
3 +1
1 +12
=1
3 +23
=311
Continued Fractions
Continued fraction expansion of ab is a path from∞ to a
b !
311 =
1
3 +1
1 +1
2 +1∞
But wait, there’s more!
New model shows there are hidden cells in our topograph livingoff near the horizon.
√2 = 1+
1
2 +1
2 +1
2 +1. . .
Back to the River
f(x, y) = x2 − 2y2 =⇒ fy2 =
(xy
)2− 2
Lagrange: If d ≥ 0 and a, b, d are integers, then the continuedfraction expansion of a + b
√d is eventually periodic.
Topographical Trinitarianism
The End
Want More?
The Sensual (Quadratic) Formby John H. Conway.
Topology of Numbersby Allen Hatcher.
