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On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic

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on-vanishing-properties-of-polynomials-on-symmetric-sets-of-the-boolean-2Representation Theorymath.COmath.RTposed by Srikanth Srinivasan, S. Venkiteshrecorded: open · 1 machine check, unexamined

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Statement

Let d∈[0,n]d \in [0, n] and E⊆[0,n]E \subseteq [0, n]. For j∈[0,n]j \in [0, n], if j∉Z-cln,d(E)j \notin Z\text{-cl}_{n,d}(E), then there exists a polynomial P(X)=ℓ1(X)⋯ℓk(X)σ(X)∈Fp[X]P(\mathbb{X}) = \ell_1(\mathbb{X}) \cdots \ell_k(\mathbb{X})\sigma(\mathbb{X}) \in \mathbb{F}_p[\mathbb{X}], where deg⁡ℓ1=⋯=deg⁡ℓk=1\deg \ell_1 = \cdots = \deg \ell_k = 1, and σ(X)\sigma(\mathbb{X}) is a symmetric polynomial, such that deg⁡P≤d\deg P \le d, P∣i‾=0P|_{\underline{i}} = 0 and P∣j‾≠0P|_{\underline{j}} \neq 0.

Context

Candidate 2 of the open problems stated in "On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Conjecture 4.5 is read as follows. Over Fp\mathbb F_p, for E⊆[0,n]E\subseteq[0,n], write

    E‾={x∈{0,1}n:∣x∣∈E}.\underline E=\{x\in\{0,1\}^n: |x|\in E\}.

    If j∉Z-cln,d(E)j\notin \mathrm{Z\text{-}cl}_{n,d}(E), then there should exist

    P(X)=ℓ1(X)⋯ℓk(X)σ(X)∈Fp[X1,…,Xn],P(X)=\ell_1(X)\cdots \ell_k(X)\sigma(X)\in\mathbb F_p[X_1,\dots,X_n],

    with each ℓr\ell_r of degree 11, σ\sigma symmetric, deg⁡P≤d\deg P\le d, P∣E‾=0P|_{\underline E}=0, and P∣j‾≠0P|_{\underline j}\ne0. The quoted P∣i‾=0P|_{\underline i}=0 has ii unbound, so P∣E‾=0P|_{\underline E}=0 is the natural intended reading.

    Result: The conjecture is false.

    Take p=2p=2, n=6n=6, d=2d=2, E={0,3,6}E=\{0,3,6\}, and j=1j=1. Let

    F=∑i=16Xi+∑1≤a<b≤5XaXb∈F2[X1,…,X6].F=\sum_{i=1}^6 X_i+\sum_{1\le a<b\le 5}X_aX_b\in\mathbb F_2[X_1,\dots,X_6].

    If ∣x∣=0|x|=0 or 66, then F(x)=0F(x)=0. If ∣x∣=3|x|=3, writing t=x6t=x_6, the first five coordinates have weight 3−t3-t, so

    F(x)=3+(3−t2)≡0(mod2).F(x)=3+\binom{3-t}{2}\equiv 0 \pmod 2.

    But if ∣x∣=1|x|=1, then F(x)=1F(x)=1. Hence 1∉Z-cl6,2({0,3,6})1\notin \mathrm{Z\text{-}cl}_{6,2}(\{0,3,6\}).

    Now suppose a conjectured witness P=GσP=G\sigma existed, where GG is a product of affine linear forms and σ\sigma is symmetric. Since deg⁡P≤2\deg P\le2, write

    P=c+∑iaiXi+∑i<jbijXiXj.P=c+\sum_i a_iX_i+\sum_{i<j}b_{ij}X_iX_j.

    From P(0)=0P(0)=0, c=0c=0. Since PP vanishes on every weight-33 vector, for every triple TT,

    ∑i∈Tai+∑{i,j}⊆Tbij=0.\sum_{i\in T}a_i+\sum_{\{i,j\}\subseteq T}b_{ij}=0.

    Summing these equations over the four triples inside any 44-set gives ∑i∈Sai=0\sum_{i\in S}a_i=0. Comparing two such 44-sets differing in one index shows all aia_i are equal. Since P∣1‾≠0P|_{\underline1}\ne0, all ai=1a_i=1. Thus P(ei)=1P(e_i)=1 for every unit vector eie_i.

    Over F2\mathbb F_2, a product of affine linear forms is the indicator of an affine subspace AA. Since σ\sigma is constant on 1‾\underline1, the equality P(ei)=1P(e_i)=1 for all ii forces ei∈Ae_i\in A for all ii. Hence AA contains the affine span of the unit vectors, namely the odd-weight hyperplane, so AA is either the whole cube or the odd-weight hyperplane.

    If AA is the whole cube, then PP is symmetric of degree at most 22, so on weight ww,

    P=α0+α1(w1)+α2(w2).P=\alpha_0+\alpha_1\binom w1+\alpha_2\binom w2.

    Vanishing at weights 0,3,60,3,6 forces α0=α1=α2=0\alpha_0=\alpha_1=\alpha_2=0, contradicting P∣1‾≠0P|_{\underline1}\ne0.

    If AA is the odd-weight hyperplane, then PP vanishes on all weight-22 vectors. But P(ei+ej)=1+1+bij=bijP(e_i+e_j)=1+1+b_{ij}=b_{ij}, so all bij=0b_{ij}=0. Then on any weight-33 vector, P=1+1+1=1P=1+1+1=1, contradiction.

    Therefore no witness of the conjectured form exists.

    Citation: Conjecture source: Srikanth Srinivasan and S. Venkitesh, “On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic,” arXiv:2111.05445, Conjecture 4.5. The counterexample above is self-contained.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The counterexample is mathematically sound and attacks Conjecture 4.5. The displayed degree-2 polynomial proves 1∉Z-cl6,2({0,3,6})1\notin Z\text{-cl}_{6,2}(\{0,3,6\}). The no-witness argument is rigorous: any quadratic vanishing on layers 00 and 33 and nonzero on layer 11 must be 11 on all unit vectors; the support of a product of affine linear forms over F2\mathbb F_2 is an affine subspace containing all unit vectors, hence either the whole cube or the odd-weight hyperplane, and both cases contradict the required vanishing. I found no existing resolution of this weaker Conjecture 4.5 in the accessible arXiv/source searches; the paper’s known counterexample is only to the earlier stronger generalized-monomial conjecture.

      Novelty assessment

      TYPE1

      Classification rationale: The counterexample is genuinely useful but very small and highly targeted: it refutes one specialized conjecture from a recent arXiv paper by a concrete p=2,n=6,d=2p=2,n=6,d=2 example. The argument is short and self-contained, so it is more appropriate as a note/corrigendum or an observation to communicate to the authors than as a standalone journal paper.

      Literature check: I found no prior publication or note containing this counterexample or an equivalent refutation of Conjecture 4.5. The original arXiv record has only v1 and no journal update. Semantic Scholar lists only three citing works; the later relevant ones concern affine Hilbert functions or real hyperplane/polynomial covering problems, not this positive-characteristic structured-witness conjecture. In particular, Ghosh–Kayal–Nandi–Venkitesh (2023) cites the Srinivasan–Venkitesh paper only as related finite-degree Zariski-closure work and solves different real hyperplane-cover questions.

      I also searched exact phrases and parameters including “Conjecture 4.5” “Z-cl”, “finite-degree Zariski closure” “Boolean cube”, the paper title, and the specific {0,3,6}\{0,3,6\} example across web search, Semantic Scholar, arXiv-accessible records, alphaXiv/SciRate, and GitHub repositories/issues/discussions. I found no matching resolution or counterexample.

      Citation: Srikanth Srinivasan and S. Venkitesh, “On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic,” arXiv:2111.05445, Conjecture 4.5. Related but non-resolving: Arijit Ghosh, Chandrima Kayal, Soumi Nandi, and S. Venkitesh, arXiv:2307.16881.

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