A Three-Polynomial Counterexample to the Generalized Effective Hypercube Nullstellensatz
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…
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