On a conjecture of C. Berenstein and A. Yger

Abstract

Let d ≥ 5 and k ≥ 1 be two integers, and let n = 10k + 1. A well-known example of E. Mayr and A. Meyer (see [MM]) shows that there are n polynomials f ₁,…,f _n Є C[x ₁,…, x _n ] of degree ≤ d such that x ₁ belongs to the ideal generated by f ₁,…, f _n that and each solution a ₁,…, a _n of the equation

$$\mathop x\nolimits_1 = \mathop a\nolimits_1 \mathop f\nolimits_1 + ... + \mathop a\nolimits_n \mathop f\nolimits_n ,{\rm }\mathop a\nolimits_{1,...,} \mathop a\nolimits_n \in C\left[ {\mathop x\nolimits_1 ,....\mathop x\nolimits_n } \right]$$

satisfies max deg $\mathop a\nolimits_i > \mathop {(d - 2)}\nolimits^{\mathop 2\nolimits^{k - 1} }$. In other words, the growth of the degrees of the polynomial coefficients in the representation problem for an ideal a $\subseteq C\left[ {\mathop x\nolimits_1 ,...,\mathop x\nolimits_n } \right]$ is, in general, double-exponential.