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Register a new userOn the linear independence of shifted powers
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Added by
Ignacio Garc\\'ia-Marco
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by $F$. In particular, we give simple criteria ensuring that the dimension of the span of $F$ is at least $c.|F|$ for some absolute constant $c<1$. We also propose conjectures implying the linear independence of the elements of $F$. These conjectures are known to be true for the field of real numbers, but not for the field of complex numbers.
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Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $phi$ be the canonical map from $Hom(V,W)$ to $Hom(wedge^t V,wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of $phi$. In the case $t=min(m,n)$, $Y_t=X_t$ is the cone over a Grassmannian, but $X_t$ is larger than $Y_t$ for $1<t<min(m,n)$. We analyze the $G=GL(V)timesGL(W)$-orbits in $X_t$ via the corresponding $G$-stable prime ideals. It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in $X_tsetminus Y_t$ arise from the images $Y_u$ for $u<t$ and simple algebraic operations. In the last section we determine the singular locus of $X_t$. Apart from well-understood exceptional cases, it is formed by the elements of rank $le 1$ in $Y_t$.
The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see, this condition is strongly correlated to good primary decompositions of the products and good homological and arithmetical properties of the associated multi-Rees algebras. The following families will be discussed in detail: polymatroidal ideals, ideals generated by linear forms and Borel fixed ideals of maximal minors. The main tools are Grobner bases and Sagbi deformation.
The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal $I$ is strictly increasing if $I$ has linear relations. Examples show that this need not to be the case for monomial ideals generated in degree greater than two.
Let $ A subset B$ be rings. An ideal $ J subset B$ is called power stable in $A$ if $ J^n cap A = (Jcap A)^n$ for all $ ngeq 1$. Further, $J$ is called ultimately power stable in $A$ if $ J^n cap A = (Jcap A)^n$ for all $n$ large i.e., $ n gg 0$. In this note, our focus is to study these concepts for pair of rings $ R subset R[X]$ where $R$ is an integral domain. Some of the results we prove are: A maximal ideal $textbf{m}$ in $R[X]$ is power stable in $R$ if and only if $ wp^t $ is $ wp-$primary for all $ t geq 1$ for the prime ideal $wp = textbf{m}cap R$. We use this to prove that for a Hilbert domain $R$, any radical ideal in $R[X]$ which is a finite intersection of G-ideals is power stable in $R$. Further, we prove that if $R$ is a Noetherian integral domain of dimension 1 then any radical ideal in $R[X] $ is power stable in $R$, and if every ideal in $R[X]$ is power stable in $R$ then $R$ is a field. We also show that if $ A subset B$ are Noetherian rings, and $ I $ is an ideal in $B$ which is ultimately power stable in $A$, then if $ I cap A = J$ is a radical ideal generated by a regular $A$-sequence, it is power stable. Finally, we give a relationship in power stability and ultimate power stability using the concept of reduction of an ideal (Theorem 3.22).
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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