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Register a new userAlexander duality for the alternative polarizations of strongly stable ideals
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We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I subset Bbbk[x_1, ldots, x_n]$ with ${rm deg}(mathsf{m}) le d$ for all $mathsf{m} in G(I)$, its dual $I^* subset Bbbk[y_1, ldots, y_d]$ is a strongly stable ideal with ${rm deg}(mathsf{m}) le n$ for all $mathsf{m} in G(I^*)$. This duality has been constructed by Fl$o$ystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert serieses of the local cohomologies $H_mathfrak{m}^i(S/I)$ using the irreducible decomposition of $I$ (through the Betti numbers of $I^*$).
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For a pair $(P,Q)$ of finite posets the generators of the ideal $L(P,Q)$ correspond bijectively to the isotone maps from $P$ to $Q$. In this note we determine all pairs $(P,Q)$ for which the Alexander dual of $L(P,Q)$ coincides with $L(Q,P)$, up to a switch of the indices.
We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $tin {1,2}$.
Let $K$ be a field and let $S=K[x_1,dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, given some positive integers $a_1,dots,a_r$ and some pairs of positive integers $(k_1,ell_1),dots,(k_r,ell_r)$, we are able to determine under which conditions there exist a $t$-spread strongly stable ideal $I$ of $S$ with $beta_{k_i, k_iell_i}(I)=a_i$, $i=1, ldots, r$, as extremal Betti numbers, and then to construct it.
Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(widetilde X)$.
We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ tgeq 1 $ for the prime ideal $ P = M cap R $. Using this we prove that for a Hilbert domain $R$ any radical ideal in $R[X]$ which is a finite intersection G-ideals is power stable. Further, we prove that if $ R $ is a Noetherian integral domain of dimension 1 then any radical ideal in $ R[X] $ is power stable. Finally, it is proved that if every ideal in $ R[X]$ is power stable then $ R $ is a field.
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