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Extensions of the notions of polynomial and rational hull
Alexander J. Izzo
Banach Algebras and Applications, July 2019, University of Manitoba
Outline
I. Polynomial and Rational Hulls
II. Motivating Questions
III. The New Hulls
IV. Applications
Polynomial and Rational Convexity
X ⊂ Cn compact
Definition: The polynomial hull of X ⊂ Cn is the set
X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X
|p(x)| for every polynomial p}.
X is said to be polynomially convex if X̂ = X .
X̂ is said to be nontrivial if X̂ \ X 6= ∅.
Polynomial and Rational Convexity
X ⊂ Cn compact
Definition: The polynomial hull of X ⊂ Cn is the set
X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X
|p(x)| for every polynomial p}.
X is said to be polynomially convex if X̂ = X .
X̂ is said to be nontrivial if X̂ \ X 6= ∅.
P (X)= uniform closure of polynomials in z1, . . . , zn on X
X̂ is the maximal ideal space of P (X).
In particular, P (X) = C(X) =⇒ X̂ = X .
Examples
In the plane, a compact set is polynomially convex if and only its com-
plement is connected. The polynomial hull of a compact set in the plane
is obtained by filling in the holes.
Examples
In the plane, a compact set is polynomially convex if and only its com-
plement is connected. The polynomial hull of a compact set in the plane
is obtained by filling in the holes.
The situation is vastly more complicated in CN for N ≥ 2. Consider
K1 = {(eiθ, 0) : 0 ≤ θ ≤ 2π} K2 = {(e
iθ, e−iθ) : 0 ≤ θ ≤ 2π}
Both are circles, but K̂1 is the disc {(z, 0) : |z| ≤ 1}, while K2 is poly-
nomially convex.
Examples
In the plane, a compact set is polynomially convex if and only its com-
plement is connected. The polynomial hull of a compact set in the plane
is obtained by filling in the holes.
The situation is vastly more complicated in CN for N ≥ 2. Consider
K1 = {(eiθ, 0) : 0 ≤ θ ≤ 2π} K2 = {(e
iθ, e−iθ) : 0 ≤ θ ≤ 2π}
Both are circles, but K̂1 is the disc {(z, 0) : |z| ≤ 1}, while K2 is poly-
nomially convex.
There exist non-polynomially convex arcs (Wermer 1955) and Cantor
sets (Rudin 1956).
Polynomial and Rational Convexity
X ⊂ Cn compact
Definition: The polynomial hull of X ⊂ Cn is the set
X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X
|p(x)| for every polynomial p}.
Polynomial and Rational Convexity
X ⊂ Cn compact
Definition: The polynomial hull of X ⊂ Cn is the set
X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X
|p(x)| for every polynomial p}.
Definition: The rational hull of X ⊂ Cn is the set
hr(X) = {z ∈ CN : p(z) ∈ p(X) for all polynomials p}.
X is said to be rationally convex if hr(X) = X .
hr(X) is said to be nontrivial if hr(X) \ X 6= ∅.
Polynomial and Rational Convexity
X ⊂ Cn compact
Definition: The polynomial hull of X ⊂ Cn is the set
X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X
|p(x)| for every polynomial p}.
Definition: The rational hull of X ⊂ Cn is the set
hr(X) = {z ∈ CN : p(z) ∈ p(X) for all polynomials p}.
X is said to be rationally convex if hr(X) = X .
hr(X) is said to be nontrivial if hr(X) \ X 6= ∅.
R(X)= uniform closure of rational functions holomorphic on X
hr(X) is the maximal ideal space of R(X).
In particular, R(X) = C(X) =⇒ hr(X) = X .
Analytic Structure in Polynomial Hulls
Observe: If X ⊂ Cn bounds an analytic variety V , then by the maximum
principle, V is contained in X̂.
Analytic Structure in Polynomial Hulls
Observe: If X ⊂ Cn bounds an analytic variety V , then by the maximum
principle, V is contained in X̂.
“Conjecture”: Every nontrivial polynomial hull contains an analytic disc.
Analytic Structure in Polynomial Hulls
Observe: If X ⊂ Cn bounds an analytic variety V , then by the maximum
principle, V is contained in X̂.
“Conjecture”: Every nontrivial polynomial hull contains an analytic disc.
Wermer (1958) prove this in the case of real-analytic 1-manifolds.
Analytic Structure in Polynomial Hulls
Observe: If X ⊂ Cn bounds an analytic variety V , then by the maximum
principle, V is contained in X̂.
“Conjecture”: Every nontrivial polynomial hull contains an analytic disc.
Wermer (1958) prove this in the case of real-analytic 1-manifolds.
Theorem (Alexander 1971): If J is a rectifiable arc in Cn, then J is
polynomially convex (and P (J) = C(J)).
Theorem (Alexander 1971): If J is a rectifiable simple closed curve in
Cn, then either J is polynomially convex (and P (J) = C(J)) or else
Ĵ \ J a one-dimensional complex analytic subvariety of Cn \ J .
Polynomial Hulls without Analytic Discs
Theorem (Stolzenberg 1963): There exists a compact set X in C2 such
that X̂ \ X 6= ∅ but X̂ contains no analytic discs.
Polynomial Hulls without Analytic Discs
Theorem (Stolzenberg 1963): There exists a compact set X in C2 such
that X̂ \ X 6= ∅ but X̂ contains no analytic discs.
Many later examples:
Wermer (1970, 1982)
Duval and Levenberg (1997)
Alexander (1998)
Polynomial Hulls without Analytic Discs
Theorem (Stolzenberg 1963): There exists a compact set X in C2 such
that X̂ \ X 6= ∅ but X̂ contains no analytic discs.
Many later examples:
Wermer (1970, 1982)
Duval and Levenberg (1997)
Alexander (1998)
Polynomial Hulls with Dense Invertibles
Theorem (Dales and Feinstein 2008): There exists a compact set X in
C2 with X̂ \ X 6= ∅ such that P (X) has dense invertibles.
Topology of Sets with Hull without Analytic Discs
Theorem (I., Samuelsson Kalm, Wold, 2016; I., Stout 2018): Every
smooth compact manifold of real dimension ≥ 2 smoothly embeds in
CN for some N so as to have nontrivial polynomial hull without analytic
discs.
Topology of Sets with Hull without Analytic Discs
Theorem (I., Samuelsson Kalm, Wold, 2016; I., Stout 2018): Every
smooth compact manifold of real dimension ≥ 2 smoothly embeds in
CN for some N so as to have nontrivial polynomial hull without analytic
discs.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
Topology of Sets with Hull without Analytic Discs
Theorem (I., Samuelsson Kalm, Wold, 2016; I., Stout 2018): Every
smooth compact manifold of real dimension ≥ 2 smoothly embeds in
CN for some N so as to have nontrivial polynomial hull without analytic
discs.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
A similar question was raised by Wermer (1954) but for 1-dimensional
manifold in C2.
Topology of Sets with Hull without Analytic Discs
Theorem (I., Samuelsson Kalm, Wold, 2016; I., Stout 2018): Every
smooth compact manifold of real dimension ≥ 2 smoothly embeds in
CN for some N so as to have nontrivial polynomial hull without analytic
discs.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
A similar question was raised by Wermer (1954) but for 1-dimensional
manifold in C2.
Question: Which compact spaces can be embedded in some CN so as
to have nontrivial polynomial hull without analytic discs?
Topology of Sets with Hull without Analytic Discs
Theorem (I., Samuelsson Kalm, Wold, 2016; I., Stout 2018): Every
smooth compact manifold of real dimension ≥ 2 smoothly embeds in
CN for some N so as to have nontrivial polynomial hull without analytic
discs.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
A similar question was raised by Wermer (1954) but for 1-dimensional
manifold in C2.
Question: Which compact spaces can be embedded in some CN so as
to have nontrivial polynomial hull without analytic discs?
Fundamental Question: Does there exist a Cantor set with nontrivial
polynomial hull without analytic discs?
Cantor Sets with Hull with Interior
Theorem (Vitushkin 1973): There exists a Cantor set in C2 whose
polynomial hull contains an open set of C2.
Cantor Sets with Hull with Interior
Theorem (Vitushkin 1973): There exists a Cantor set in C2 whose
polynomial hull contains an open set of C2.
Theorem (Henkin 2006): There exists a Cantor set in C2 whose rational
hull contains an open set of C2.
Cantor Sets with Hull with Interior
Theorem (Vitushkin 1973): There exists a Cantor set in C2 whose
polynomial hull contains an open set of C2.
Theorem (Henkin 2006): There exists a Cantor set in C2 whose rational
hull contains an open set of C2.
Question: Can Henkin’s theorem be generalized to CN for N > 2?
Cantor Sets with Hull with Interior
Theorem (Vitushkin 1973): There exists a Cantor set in C2 whose
polynomial hull contains an open set of C2.
Theorem (Henkin 2006): There exists a Cantor set in C2 whose rational
hull contains an open set of C2.
Question: Can Henkin’s theorem be generalized to CN for N > 2?
Answer: Yes, but a direct generalization is not so interesting, because
while in C2 to say z ∈ hr(X) means every analytic variety through z
intersects X , in contrast, in CN , N > 2, hr(X) concerns only codimen-
sion 1 analytic varieties.
k-Hulls
Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set
hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure
codimension ≤ k that passes through z intersects X}.
We say that X is k-rationally convex if hkr (X) = X .
k-Hulls
Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set
hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure
codimension ≤ k that passes through z intersects X}.
We say that X is k-rationally convex if hkr (X) = X .
How to define k-polynomial hull?
k-Hulls
Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set
hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure
codimension ≤ k that passes through z intersects X}.
We say that X is k-rationally convex if hkr (X) = X .
How to define k-polynomial hull?
Note: If z ∈ h2r(X), then for V = {p = 0} (p a polynomial), z ∈
hr((X ∩ V )).
k-Hulls
Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set
hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure
codimension ≤ k that passes through z intersects X}.
We say that X is k-rationally convex if hkr (X) = X .
How to define k-polynomial hull?
Note: If z ∈ h2r(X), then for V = {p = 0} (p a polynomial), z ∈
hr((X ∩ V )).
To define 2-polynomial hull, replace z ∈ hr((X ∩ V )) by z ∈ X̂ ∩ V .
k-Hulls
Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set
hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure
codimension ≤ k that passes through z intersects X}.
We say that X is k-rationally convex if hkr (X) = X .
Definition: For 2 ≤ k ≤ N , the k-polynomial hull X̂k of X is the set
X̂k = {z ∈ CN : z ∈ hk−1r (X) and z ∈ X̂ ∩ V for every analytic subvariety V
of CN of pure codimension ≤ k − 1 that passes through z}.
We say that X is k-polynomially convex if X̂k = X .
k-Hulls
Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set
hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure
codimension ≤ k that passes through z intersects X}.
We say that X is k-rationally convex if hkr (X) = X .
Definition: For 2 ≤ k ≤ N , the k-polynomial hull X̂k of X is the set
X̂k = {z ∈ CN : z ∈ hk−1r (X) and z ∈ X̂ ∩ V for every analytic subvariety V
of CN of pure codimension ≤ k − 1 that passes through z}.
We say that X is k-polynomially convex if X̂k = X .
With these definitions
X̂ = X̂1 ⊃ hr(X) = h1
r(X) ⊃ X̂2 ⊃ h2r(X) ⊃ · · · ⊃ X̂
n ⊃ hnr (X) = X.
Hulls without Analytic Discs
Three Fundamental Constructions
(i) Stolzenberg 1963: Take a limit of boundaries of analytic varieties whose
hulls are such that their projections to the coordinate planes miss points
of a dense set.
(ii) Wermer 1970: Successively remove sets from the boundary of a domain
in CN in such a way that what is left in the limit has hull without analytic
discs.
(iii) Wermer 1982 (based on Cole 1968): Take a limit of graphs of multivalued
analytic functions involving square roots to get a “Riemann surface with
an infinite number of branch points”.
Hulls without Analytic Discs
Fundamental Construction (ii)
(ii) Wermer 1970: Successively remove sets from the boundary of a domain
in CN in such a way that what is left in the limit has hull without analytic
discs.
Hulls without Analytic Discs
Fundamental Construction (ii)
(ii) Wermer 1970: Successively remove sets from the boundary of a domain
in CN in such a way that what is left in the limit has hull without analytic
discs.
The key to (ii) is to cut out subsets with the property that a point of
the domain lies in the rational hull of the set that remains if it does not
lie in the polynomial hull of the set removed. One approach uses:
Hulls without Analytic Discs
Fundamental Construction (ii)
(ii) Wermer 1970: Successively remove sets from the boundary of a domain
in CN in such a way that what is left in the limit has hull without analytic
discs.
The key to (ii) is to cut out subsets with the property that a point of
the domain lies in the rational hull of the set that remains if it does not
lie in the polynomial hull of the set removed. One approach uses:
Lemma: Let p be a polynomial on CN and X = {ℜp ≤ 0} ∩ ∂B. (B =
unit ball in CN ) Then X̂ = hr(X) = {ℜp ≤ 0} ∩ B.
Hulls without Analytic Discs
Fundamental Construction (ii)
(ii) Wermer 1970: Successively remove sets from the boundary of a domain
in CN in such a way that what is left in the limit has hull without analytic
discs.
The key to (ii) is to cut out subsets with the property that a point of
the domain lies in the rational hull of the set that remains if it does not
lie in the polynomial hull of the set removed. One approach uses:
Lemma: Let p be a polynomial on CN and X = {ℜp ≤ 0} ∩ ∂B. (B =
unit ball in CN ) Then X̂ = hr(X) = {ℜp ≤ 0} ∩ B.
What makes the proof of the lemma work is that ∂̂B2
= B.
Hulls without Analytic Discs
Fundamental Construction (ii)
(ii) Wermer 1970: Successively remove sets from the boundary of a domain
in CN in such a way that what is left in the limit has hull without analytic
discs.
The key to (ii) is to cut out subsets with the property that a point of
the domain lies in the rational hull of the set that remains if it does not
lie in the polynomial hull of the set removed. One approach uses:
Lemma: Let p be a polynomial on CN and X = {ℜp ≤ 0} ∩ ∂B. (B =
unit ball in CN ) Then X̂ = hr(X) = {ℜp ≤ 0} ∩ B.
What makes the proof of the lemma work is that ∂̂B2
= B.
With this observation one can generalize the lemma to get a very flexible
method of constructing hulls without analytic discs.
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Corollary: For N ≥ 2, there exists a totally disconnected, perfect set in
CN that intersects every analytic subvariety of CN of positive dimension.
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Theorem: For N ≥ 3, there exists a Cantor set K in CN with nontrivial
(N−2)-rational hull and whose polynomial hull contains no analytic discs
(and such that P (K) has dense invertibles).
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Theorem: For N ≥ 3, there exists a Cantor set K in CN with nontrivial
(N−2)-rational hull and whose polynomial hull contains no analytic discs
(and such that P (K) has dense invertibles).
Corollary: There exists a simple closed curve (and an arc) in C3 with
nontrivial polynomial hull containing no analytic discs.
Applications
Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull
(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such
that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Theorem: For N ≥ 3, there exists a Cantor set K in CN with nontrivial
(N−2)-rational hull and whose polynomial hull contains no analytic discs
(and such that P (K) has dense invertibles).
Corollary: There exists a simple closed curve (and an arc) in C3 with
nontrivial polynomial hull containing no analytic discs.
Corollary: Every uncountable, compact, metrizable space of finite topo-
logical dimension can be embedded in some CN so as to have nontrivial
polynomial hull without analytic discs.
Open Question
Does there exist a Cantor set in C2 with a nontrivial polynomial hull
that contains no analytic discs?