Asymptotic behaviour of the v-number of homogeneous ideals

Abstract: Let S = K[x1,...,xn] be the standard graded polynomial with coefficients over a field K, and let I ⊂ S be a homogeneous ideal. The v-number of I is defined as the minimum degree of an homogeneous polynomial f ∈ S such that (I : f) ∈ Ass(I) is an associated prime of I. This invariant was introduced in relation to minimum distance functions and Reed-Muller type codes. In the present talk, we show that the function v(Ik) is an eventually linear function α(I)k + b, where α(I) is the initial degree of I and b is a suitable integer. We then survey the recent numerous studies on this and related topics, and some open questions.

Keywords: Associated primes. v-number. Graded rings. Monomial ideals. Integer Programming.