The v-number of monomial ideals

摘要

We show that the v-number of an arbitrary monomial ideal is bounded below by the v-number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the v-number v(I(G)) of the edge ideal I (G), the induced matching number im(G) and the regularity reg(R /I (G)) of a graph G, satisfy v(I (G)) reg(R / I (G)) + 1, for a disconnected graph G. We derive some inequalities of v-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v (I (G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg(R / I (G)) can be arbitrarily larger than v(I (G)). Also, we try to see how the v-number is related to the Cohen-Macaulay property of square-free monomial ideals.