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We study the number of generators of ideals in regular rings and ask the question whether $mu(I)<mu(I^2)$ if $I$ is not a principal ideal, where $mu(J)$ denotes the number of generators of an ideal $J$. We provide lower bounds for the number of generators for the powers of an ideal and also show that the CM-type of $I^2$ is $geq 3$ if $I$ is a monomial ideal of height $n$ in $K[x_1,ldots,x_n]$ and $ngeq 3$.
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Given a number $q$, we construct a monomial ideal $I$ with the property that the function which describes the number of generators of $I^k$ has at least $q$ local maxima.
Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${rm reg}(I^{(s)}) = {rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds begin{align*} {rm reg} I^{s} & le {rm reg} I + 2s - 2, text{ for } s = 2,3, {rm reg} I^{(s)} & le {rm reg} I + 2s - 2,text{ for } s = 2,3,4. end{align*}
We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal.
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $mu (I^2)ge 9$ for the number of minimal generators of $I^2$ with $mu(I)geq 6$. Recently, Gasanova constructed monomial ideals such that $mu(I)>mu(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $mu(I)>mu(I^2)>cdots >mu(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $mu(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends.
Let $A={{bf a}_1,...,{bf a}_m} subset mathbb{Z}^n$ be a vector configuration and $I_A subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of $I_A$. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to $A$ a simplicial complex $Delta _{ind(A)}$. We show that the vertices of $Delta_{ind(A)}$ correspond to the indispensable monomials of the toric ideal $I_A$, while one dimensional facets of $Delta_{ind(A)}$ with minimal binomial $A$-degree correspond to the indispensable binomials of $I_{A}$.
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