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A Three-Polynomial Counterexample to the Generalized Effective Hypercube Nullstellensatz

Primarily AI-generated textHuman understanding: no partscs.CC — Computational Complexity

Contributed by YURAN XIU ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

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≥1n\geq1, we construct three real polynomials in 2n+12n+1 variables, of degrees 2,2,12,2,1, 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 ∑i=13higi=1\sum_{i=1}^3 h_i g_i=1 on the cube satisfies deg⁡h1g1‾≥n+1\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.

References

  1. Robin Kothari, Matt Kovacs-Deak, Daochen Wang, and Rain Zimin Yang. Rational degree is polynomially related to degree. arXiv:2601.08727v3 (https://arxiv.org/abs/2601.08727v3), 28 September 2026. Section 5.1, Theorem 10, Conjecture 1, and Fact 7.arXiv

Version history

  1. v1Submitted by YURAN XIUInitial depositCurrentOct 18, 2026