Hexagon

cs.CC — Computational Complexity

A Three-Polynomial Counterexample to the Generalized Effective Hypercube Nullstellensatz

Contributed by YURAN XIU

We give a counterexample to the proposed extension of the effective Hypercube Nullstellensatz from two polynomials to an arbitrary fixed number of polynomials. For every $n\geq1$ , we construct three real polynomials in $2n+1$ variables, of degrees $2,2,1$ , respectively. Each is nonzero as a function on the Boolean cube, their product vanishes on the cube, and they have no common Boolean zero. Nevertheless, every polynomial certificate $\sum_{i=1}^3 h_i g_i=1$ on the cube satisfies $\deg\overline{h_1g_1}\geq n+1$ , where the bar denotes multilinearization. The construction applies a fresh selector variable to the hard pair of Kothari, Kovacs-Deak, Wang, and Yang. Thus its new ingredient is a short transfer of their existing degree obstruction to the product-zero setting. The full unbounded-degree counterexample has been formalized and checked in Lean 4. HAL deposit: hal-05800244v1.

Primarily AI-generated textHuman understanding: no partsBoolean cubeLean formalizationNullstellensatzdegree lower boundspolynomial certificates

Oct 18, 2026PDF ↗View work →

hexagon:2610.00328v1

cs.CC — Computational Complexity

A Counterexample to Affine-Subspace Union Stability under Small Deletions

Contributed by YURAN XIU

We construct a family of affine coordinate subspaces of $\mathbb{F}_2^N$ whose union is the entire space, but for which deleting a $1/N$ fraction from each member removes a fraction larger than $N^{-c}$ from the union, for every fixed $c>0$ and arbitrarily large $N$ . Each subspace has codimension $\log_2 N$ , and the family has at most $2N^2$ members. The construction uses all monochromatic coordinate conditions of size $k$ on $2k$ distinguished coordinates, where $N=2^k$ , and deletes their common intersection with the middle Hamming layer. The exact global loss is $\binom{2k}{k}/4^k$ . This disproves the affine-union conjecture of Alekseev and Gaevoy, already for their fixed parameters $q=2$ and $r=1$ . The all-exponent, arbitrarily-large-dimension counterexample has been formalized and checked in Lean 4. HAL deposit: hal-05800246v1.

Primarily AI-generated textHuman understanding: no partsBoolean cubeLean formalizationaffine subspacescounterexampleunion stability

Oct 18, 2026PDF ↗View work →

hexagon:2610.00307v1

cs.CC — Computational Complexity

Set Cover by Cyclic Rotations Is NP-Complete

Contributed by YURAN XIU

Given an explicitly listed binary pattern of positive length $n$ and a nonnegative integer $b$ in binary, can at most $b$ cyclic rotations of the pattern cover all $n$ positions? We prove that this decision problem is NP-complete. The reduction constructs a pattern in a cyclic group of polynomial order and outputs its entire indicator word. Its principal ingredient is a replicated-marker construction: a cover within the budget must choose one translate in every layer, and each failed difference constraint leaves a block of more holes than the remaining translates can cover. Combining this construction with the known embedding of graph cliques into circulant graphs using distinct three-term sums gives a polynomial-time reduction from $3$ SAT. The theorem allows every subset, including the empty pattern, and every binary budget. A Lean formalization checks the mathematical reduction, the full-word polynomial-time programs, NP membership, and the unconditional NP-hardness foundation. HAL deposit: hal-05800233v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationNP-completenesscyclic rotationsset cover

Oct 17, 2026PDF ↗View work →

hexagon:2610.00308v1

cs.CC — Computational Complexity

A Polynomial Separation for Adaptive Fourier-Overlap Parity Decision Trees

Contributed by YURAN XIU

Consider the parity decision tree rule that, at every nonconstant restriction, queries a nonzero character maximizing the intersection of the current Fourier support with its translate. We construct an explicit infinite family for which every adaptive policy obeying this rule has polynomial worst-case depth, although an unrestricted parity decision tree has logarithmic depth. For every $\ell\geq1$ , let $R=2^\ell$ . Our function has $n=4\ell-1+R^2\ell+R$ input bits and exactly $k=R^6-R^5+R$ nonzero Fourier coefficients. Every history-dependent maximizing policy terminates and computes the function, but has a feasible path of length at least $R>k^{1/6}$ ; an explicit unrestricted tree has depth at most $5\ell\leq\log_2 k$ . The construction places one pointer behind each cell of a finite-field affine partition, with all pointers accessing a shared data table. After any sequence of equal-answer table comparisons, exact signed Fourier coefficients force every maximizing query to compare two remaining table blocks. This invariant accommodates all tie choices and all resulting table partitions. The complete separation, including the specified polynomial-field representation and the existence of every operational policy's finite run, has been checked in Lean. The result concerns this particular support-overlap rule; it is not a lower bound against arbitrary parity decision trees. HAL deposit: hal-05800236v1.

Primarily AI-generated textHuman understanding: no partsFourier analysisLean formalizationadaptive algorithmsparity decision trees

Oct 17, 2026PDF ↗View work →

Matching works

A Three-Polynomial Counterexample to the Generalized Effective Hypercube Nullstellensatz

A Counterexample to Affine-Subspace Union Stability under Small Deletions

Set Cover by Cyclic Rotations Is NP-Complete

A Polynomial Separation for Adaptive Fourier-Overlap Parity Decision Trees

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