Resumen de A study of {\textrm{v}}-number for some monomial ideals

Prativa Biswas, Mousumi Mandal

• In this paper, we give formulas for {\textrm{v}}-number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an {\mathfrak {m}}-primary monomial ideal I\subset S=K[x_1,\ldots ,x_t], we provide an explicit expression of {\textrm{v}}-number of I, denoted by {\textrm{v}}(I), and give an upper bound of {\textrm{v}}(I) in terms of the degree of its generators. We show that for a monomial ideal I, {\textrm{v}}(I^{n+1}) is bounded above by a linear polynomial for large n and for certain classes of monomial ideals, the upper bound is achieved for all n\ge 1. For {\mathfrak {m}}-primary monomial ideal I we prove that {\textrm{v}}(I)\le {\text {reg}}(S/I) and their difference can be arbitrarily large.