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Register a new userOn the resurgence and asymptotic resurgence of homogeneous ideals
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Let $mathbb{K}$ be a field and $R = mathbb{K}[x_1, ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.
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Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.
Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted $sdepth(M)$, and conjectured that $depth(M) le sdepth(M)$ for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M = I / J$ with $J subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I_{n,d}$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1le dle n < 5d+4$, then $sdepth(I_{n,d})= floor{binom{n}{d+1}Big/binom{n}{d}}+d$, and if $dgeq 1$ and $nge 5d+4$, then $d+3le sdepth(I_{n,d}) le floor{binom{n}{d+1}Big/binom{n}{d}}+d$.
We discuss the analytic continuation of the Hadamard product of two holomorphic functions under assumptions pertaining to Ecalles Resurgence Theory, proving that if both factors are endlessly continuable with prescribed sets of singular points $A$ and $B$, then so is their Hadamard product with respect to the set ${0}cup A cdot B$. In this generalization of the classical Hadamard Theorem, all the branches of the multivalued analytic continuation of the Hadamard product are considered.
We show how the renormalons emerge from the renormalization group equation with a priori no reference to any Feynman diagrams. The proof is rather given by recasting the renormalization group equation as a resurgent equation studied in the mathematical literature, which describes a function with an infinite number of singularities in the positive axis of the Borel plane. Consistency requires a one-to-one correspondence between the existence of such kind of equation and the actual (generalized) Borel resummation of the renormalons through a one-parameter transseries. Our finding suggests how non-perturbative contributions can affect the running couplings. We also discuss these concepts within the context of gauge theories, making use of the large number of flavor expansion.
We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissiers volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the epsilon-multiplicity of a monomial ideal in terms of the volume of a region.
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