Polynomial bounds for the stability of associated primes of monomial ideals

Abstract

Let $I$ be a nonzero proper monomial ideal in ${\rm k}[x_1,\ldots,x_r]$, and let $d$ be the maximal degree of a minimal monomial generator. We prove that the index of stability of the associated primes of its ordinary powers satisfies \[ {\rm astab}(I)\le\frac{(4e)^{r-1}}4\,d^{r-1}\qquad(r\ge3), \] while ${\rm astab}(I)=1$ for $r=2$. If $I$ has $s\ge2$ minimal generators, we also obtain \[ {\rm astab}(I)\le\min\{r,s\}(s-1)d^{\min\{r-1,s-2\}}. \] The first estimate answers Hoa's question on polynomial bounds. The proofs reduce stabilization to integer optimization for a matrix whose last row records the power of the ideal. We bound the degrees of its circuits and Graver basis elements and prove proximity estimates for optimal solutions in fixed fibers. A Steinitz rearrangement and a lattice-point count in the difference body of a simplex give the generator-independent bound. The matrix estimates hold for arbitrary indexed integral configurations in a simplex and real linear objective functions. The stability bounds hold over an arbitrary field, without normality or equigeneration assumptions.