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The $\text{v}$-function of powers of sums of ideals
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abstract
Let $K$ be a field, $I\subset R=K[x_1,\dots,x_n]$ and $J\subset T=K[y_1,\dots,y_m]$ be graded ideals. Set $S=R\otimes_KT$ and let $L=IS+JS$. The behaviour of the $\text{v}$-function $\text{v}(L^k)$ in terms of the $\text{v}$-functions $\text{v}(I^k)$ and $\text{v}(J^k)$ is investigated. When $I$ and $J$ are monomial ideals, we describe $\text{v}(L^k)$, giving an explicit formula involving $\text{v}(I^k)$ and $\text{v}(J^k)$.
Forward citations
Cited by 2 Pith papers
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$\operatorname{v}$-numbers of integral closure filtrations of monomial ideals
For certain monomial ideals, the paper proposes explicit values and bounds for v-numbers of integral closure filtrations and shows they can be smaller than v-numbers of ordinary powers.
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Comparison of stability indices of powers of graded ideals
The paper proves that astab(I)=1 for every graded ideal in dimension two with vstab(I) arbitrary, and constructs ideals realizing every pair (astab(I), vstab(I))=(a,b) in higher dimension.
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