Weak Shanks Conjecture in Bidisk Hardy Spaces

• The Weak Shanks Conjecture is a hypothesis in bidisk Hardy spaces that asserted optimal polynomial approximants for zero-free functions would be zero-free, but explicit counterexamples have disproven it.
• It originated from investigations in least-squares inverse problems and cyclicity in Hardy spaces, linking polynomial approximation methods with digital filter design stability.
• Recent Fourier-analytic techniques and hypergeometric formulations have led to new counterexample families, providing a refined framework for understanding zero behavior in multivariable function theory.

Searching arXiv for recent and foundational papers on the Weak Shanks Conjecture and closely related work. The Weak Shanks Conjecture is a conjecture in the function theory of the bidisk concerning zeros of optimal polynomial approximants in $H^2(\mathbb D^2)$. In its weak or “weakest” form, it asserts that if a polynomial $f$ is nonvanishing on $\overline{\mathbb D^2}$, then the optimal polynomial approximants to $1/f$ should be zero-free in $\mathbb D^2$. A closely related formulation quoted from Delsarte–Genin–Kamp assumes only that $f$ has no zeros in $\mathbb D^2$. This conjecture arose from least-squares inverse problems for two-variable filters and from the study of cyclicity and invariant subspaces in Hardy spaces. It is now known to be false: a first disproof was given by a degree-one counterexample in 2024, and a 2025 Fourier-analytic paper embedded that example in a broader parametric family, derived explicit hypergeometric formulas for the relevant Fourier data, and produced additional counterexamples (Bénéteau et al., 2024, Geronimo et al., 21 Aug 2025).

1. Statement and mathematical setting

The ambient space is the Hardy space $H^2(\mathbb D^2)$, where $\mathbb D=\{z\in\mathbb C:|z|<1\}$. For

$g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$

the norm is

$f$0

so the monomials $f$1 form an orthonormal basis. For each $f$2, let $f$3 denote the two-variable polynomials spanned by monomials $f$4 with $f$5. Given $f$6, an optimal polynomial approximant (OPA) of degree $f$7 to $f$8 is a polynomial $f$9 minimizing

$\overline{\mathbb D^2}$0

Equivalently, $\overline{\mathbb D^2}$1 is the orthogonal projection of $\overline{\mathbb D^2}$2 onto the finite-dimensional subspace $\overline{\mathbb D^2}$3 (Bénéteau et al., 2024).

In the bidisk literature, the weak conjecture is usually stated as follows: if $\overline{\mathbb D^2}$4 is a polynomial with no zeros in $\overline{\mathbb D^2}$5, then its OPAs are zero-free in $\overline{\mathbb D^2}$6. The 2024 counterexample paper describes this as the “weakest” form, contrasting it with stronger earlier variants that did not assume $\overline{\mathbb D^2}$7 itself was zero-free in the bidisk (Bénéteau et al., 2024). A related formulation quoted in the 2025 Fourier-analytic paper is: if $\overline{\mathbb D^2}$8 is a polynomial with no zeros in $\overline{\mathbb D^2}$9, then its OPAs are zero-free in $1/f$0 (Geronimo et al., 21 Aug 2025). In the stable-polynomial terminology used in related structural work, a polynomial is stable if it has no zeros in $1/f$1 and strongly stable if it has no zeros in $1/f$2; the weak conjecture then becomes the assertion that strongly stable polynomials have OPAs that are stable (Felder, 2024).

2. Historical origin and pre-disproof structure

The conjecture belongs to the theory of least-squares inverse polynomials in two variables, with applications to recursive digital filter design. In one variable, the analogous zero-exclusion phenomenon is true: OPAs are zero-free in the closed unit disk. That one-variable fact suggested a bidisk analogue and partly explains why the conjecture persisted. The same minimization problem is also tied to cyclicity in Hardy spaces, because a function $1/f$3 is cyclic exactly when there exist polynomials $1/f$4 such that

$1/f$5

(Bénéteau et al., 2024).

Before the disproof, structural work emphasized several positive or restrictive results. One paper distinguished a strong Shanks conjecture—asserting zero-exclusion for OPAs of arbitrary polynomial $1/f$6—from the weaker form restricted to strongly stable $1/f$7, and it studied the zero geometry of OPAs through Gram matrices, reproducing kernels, and polynomial reflection (Felder, 2024). That paper also proved positive classes: if $1/f$8 for a one-variable $1/f$9 and a monomial $\mathbb D^2$0, then every OPA is zero-free in the closed bidisk, so “one-variable functions in disguise” cannot furnish counterexamples. It further derived a determinantal criterion: for $\mathbb D^2$1 zero-free in $\mathbb D^2$2, the condition that $\mathbb D^2$3 have no zeros in $\mathbb D^2$4 is equivalent to the nonvanishing of a kernel-modified Gram determinant for every $\mathbb D^2$5 (Felder, 2024).

This pre-disproof work clarified the real difficulty. The central issue was not the existence of OPAs or the linear algebra of the optimality equations, but the relation between the zero set of $\mathbb D^2$6 and the zero sets of the approximants. The determinantal formulation made that difficulty explicit, but it did not resolve the conjecture.

3. The 2024 counterexample

The first disproof exhibits the explicit function

$\mathbb D^2$7

Because $\mathbb D^2$8 on $\mathbb D^2$9, the principal branch is analytic on a neighborhood of $f$0; hence $f$1 is holomorphic and nonzero on the closed bidisk. The degree-one OPA to $f$2 has the symmetric affine form

$f$3

with $f$4, since $f$5 is symmetric in $f$6 and has real Taylor coefficients (Bénéteau et al., 2024).

The proof reduces the bidisk minimization problem to a one-variable weighted Hilbert space $f$7. Writing

$f$8

the relevant weights are

$f$9

The normal equations become

$\mathbb D^2$0

and, using $\mathbb D^2$1, reduce to

$\mathbb D^2$2

From these, the paper derives the criterion

$\mathbb D^2$3

If $\mathbb D^2$4, then the diagonal restriction $\mathbb D^2$5 changes sign on $\mathbb D^2$6, so $\mathbb D^2$7 vanishes at

$\mathbb D^2$8

(Bénéteau et al., 2024).

The verification is analytic rather than symbolic. The coefficients of

$\mathbb D^2$9

satisfy

$H^2(\mathbb D^2)$0

and the comparison $H^2(\mathbb D^2)$1 is proved by reducing it to a finite-sum inequality with explicit tail bounds. In particular, the paper states

$H^2(\mathbb D^2)$2

and obtains

$H^2(\mathbb D^2)$3

which yields $H^2(\mathbb D^2)$4 (Bénéteau et al., 2024).

Strictly speaking, this $H^2(\mathbb D^2)$5 is not a polynomial. The same paper addresses that point directly. OPAs depend continuously on $H^2(\mathbb D^2)$6, and the Taylor polynomials $H^2(\mathbb D^2)$7 converge uniformly to $H^2(\mathbb D^2)$8 on $H^2(\mathbb D^2)$9. Hence, for sufficiently large $\mathbb D=\{z\in\mathbb C:|z|<1\}$0, the polynomial $\mathbb D=\{z\in\mathbb C:|z|<1\}$1 remains zero-free on $\mathbb D=\{z\in\mathbb C:|z|<1\}$2, and the degree-one OPA to $\mathbb D=\{z\in\mathbb C:|z|<1\}$3 remains close enough to $\mathbb D=\{z\in\mathbb C:|z|<1\}$4 to retain a zero in $\mathbb D=\{z\in\mathbb C:|z|<1\}$5. This converts the analytic example into an actual polynomial counterexample satisfying the conjecture’s stated hypotheses (Bénéteau et al., 2024).

4. Fourier-analytic reformulation and the 2025 strengthening

The 2025 paper does not provide the first disproof; rather, it develops a systematic Fourier-analytic framework that produces additional counterexamples and reorganizes the mechanism behind the 2024 example. Its central family is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$6

The condition $\mathbb D=\{z\in\mathbb C:|z|<1\}$7 places the singularity set $\mathbb D=\{z\in\mathbb C:|z|<1\}$8 outside the closed bidisk, so $\mathbb D=\{z\in\mathbb C:|z|<1\}$9 is holomorphic on a neighborhood of $g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$0. The OPA problem is then encoded by the Fourier coefficients of $g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$1 on $g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$2 (Geronimo et al., 21 Aug 2025).

If

$g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$3

and

$g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$4

then the degree-$g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$5 OPA

$g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$6

satisfies the Toeplitz-like linear system

$g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$7

The paper’s main technical theorem computes the Fourier coefficients $g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$8 and $g(z_1,z_2)=\sum_{k,l\ge 0} a_{k,l} z_1^k z_2^l,$9 in closed $f$00 form. The derivation is entirely Fourier-analytic and hypergeometric: expand $f$01, multiply by its conjugate expansion, collect coefficients, and reduce the finite combinatorial sums by factorial identities and the Chu–Vandermonde formula. This converts the OPA computation into an explicit special-function problem (Geronimo et al., 21 Aug 2025).

For degree $f$02, symmetry reduces the OPA to

$f$03

Writing

$f$04

the paper gives

$f$05

$f$06

$f$07

Because $f$08, the root condition simplifies to

$f$09

so the degree-one OPA has a zero in $f$10 exactly when

$f$11

Equivalently,

$f$12

This scalar inequality is the effective counterexample criterion (Geronimo et al., 21 Aug 2025).

The same paper also derives an integral reformulation of the degree-one root condition as a triple integral inequality involving $f$13 kernels, and remarks that this form may be more conducive to analysis in $f$14. It thereby reframes the zero problem not only in Fourier and Toeplitz terms but also in integral-special-function terms (Geronimo et al., 21 Aug 2025).

5. Explicit families and numerical optimization

The Fourier-analytic framework recovers the 2024 counterexample and supplies additional parameter choices. It first reproduces the original example by taking

$f$15

for which the numerical check is

$f$16

Thus the degree-one OPA for

$f$17

has a zero in $f$18, matching the earlier disproof (Geronimo et al., 21 Aug 2025).

The paper then gives additional examples, including two rational-function families with integer $f$19. These are not themselves polynomial inputs, but, exactly as in the 2024 disproof, sufficiently high Taylor truncations yield polynomial counterexamples.

Parameters $f$20	Verified inequality	Consequence
$f$21	$f$22	Recovers the first counterexample
$f$23	$f$24	New explicit counterexample family
$f$25	$f$26	New explicit counterexample family

The same paper introduces the quotient

$f$27

The bidisk-root condition is that this quotient be $f$28. Minimizing it over $f$29 therefore seeks degree-one OPAs whose zeros lie as close to the origin as possible. Using Matlab’s fmincon, the reported minimizer is

$f$30

with minimal quotient

$f$31

For these values, the OPA is approximately

$f$32

which has a root, for instance, at

$f$33

The authors interpret this as producing roots “the closest (in $f$34, for instance) to the origin” (Geronimo et al., 21 Aug 2025).

6. Current status, significance, and terminology

The current status is definitive: the conjecture is false. What failed is the blanket assertion that zero-freeness of a polynomial $f$35 on the bidisk, or on the closed bidisk in the weakest formulation, forces zero-freeness of all OPAs to $f$36. The 2024 paper settled the problem by a degree-one counterexample, and the 2025 paper strengthened that disproof by placing the phenomenon inside a larger family

$f$37

computing the relevant Fourier coefficients explicitly, and giving a practical optimization scheme for the parameters $f$38 (Bénéteau et al., 2024, Geronimo et al., 21 Aug 2025).

A recurrent misconception in later summaries is that the Fourier-analytic paper supplies the first disproof. It does not. Its contribution is different: additional counterexamples, explicit hypergeometric formulas for the Fourier coefficients of $f$39, a scalar degree-one criterion, an integral reformulation, and a numerical search strategy. Conversely, the fact that the first example is nonpolynomial is not an obstruction to the conjecture’s disproof, because sufficiently high Taylor truncations preserve zero-freeness on $f$40 and inherit the zero-producing degree-one OPA behavior by continuity (Bénéteau et al., 2024, Geronimo et al., 21 Aug 2025).

The broader significance of the conjecture lies in two directions. Function-theoretically, it probes how orthogonal projection in $f$41 interacts with the zero geometry of multivariable holomorphic functions. From the engineering side, the problem comes from least-squares inverse polynomials in two variables, where zero-freeness of approximants corresponds to a stability property in recursive digital filter design (Bénéteau et al., 2024). The disproof therefore marks a sharp departure from the one-variable Hardy-space picture.

The phrase “Weak Shanks Conjecture” is not unique across mathematics. In unrelated arithmetic literature, one paper uses it for a conjectural statement about $f$42-Shanks primes and shows, under hypotheses including irreducibility of $f$43 modulo $f$44, that

$f$45

(Jones, 2023). That number-theoretic usage is distinct from the bidisk Hardy-space conjecture discussed here.