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Register a new userFreiman $t$-spread principal Borel ideals
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An equigenerated monomial ideal $I$ is a Freiman ideal if $mu(I^2)=ell(I)mu(I)-{ell(I)choose 2}$ where $ell(I)$ is the analytic spread of $I$ and $mu(I)$ is the least number of monomial generators of $I$. Freiman ideals are special since there exists an exact formula computing the least number of monomial generators of any of their powers. In this paper we give a complete classification of Freiman $t$-spread principal Borel ideals.
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Fix a poset $Q$ on ${x_1,ldots,x_n}$. A $Q$-Borel monomial ideal $I subseteq mathbb{K}[x_1,ldots,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a principal $Q$-Borel ideal, denoted $I=Q(m)$, if there is a monomial $m$ such that all the minimal generators of $I$ can be obtained via $Q$-Borel moves from $m$. In this paper we study powers of principal $Q$-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset $Q$.
In this paper it is shown that a sortable ideal $I$ is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals with constant bound.
Fix a square-free monomial $m in S = mathbb{K}[x_1,ldots,x_n]$. The square-free principal Borel ideal generated by $m$, denoted ${rm sfBorel}(m)$, is the ideal generated by all the square-free monomials that can be obtained via Borel moves from the monomial $m$. We give upper and lower bounds for the Waldschmidt constant of ${rm sfBorel}(m)$ in terms of the support of $m$, and in some cases, exact values. For any rational $frac{a}{b} geq 1$, we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to $frac{a}{b}$.
An equigenerated monomial ideal $I$ in the polynomial ring $R=k[z_1,ldots, z_n]$ is a Freiman ideal if $mu(I^2)=ell(I)mu(I)-{ell(I)choose 2}$ where $ell(I)$ is the analytic spread of $I$ and $mu(I)$ is the number of minimal generators of $I$. In this paper we classify all simple connected unmixed bipartite graphs whose cover ideals are Freiman ideals.
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 $tin {1,2}$.
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