arXiv subjects

Anthony V. Geramita

Publications and source records attributed to Anthony V. Geramita.

16 recordsLinked to original sources

arXiv2018

The symbolic defect of an ideal

Let $I$ be a homogeneous ideal of $\Bbbk[x_0,\ldots,x_n]$. To compare $I^{(m)}$, the $m$-th symbolic power of $I$, with $I^m$, the regular $m$-th power, we introduce the $m$-th symbolic defect of $I$, denoted $\operatorname{sdefect}(I,m)$. Precisely, $\operatorname{sdefect}(I,m)$ is the minimal number of generators of the $R$-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that $I$ is either the defining ideal of a star configuration or the ideal associated to a finite set of points in $\mathbb{P}^2$. We are specifically interested in identifying ideals $I$ with $\operatorname{sdefect}(I,2) = 1$.

math.AC↗

arXiv2017

Symmetric Complete Intersections

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals, and describe formulas for the graded characters of the corresponding quotient rings.

math.AC↗

arXiv2017

Waring-like decompositions of polynomials - 1

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.

math.AG↗

arXiv2016

Degrees of regular sequences with a symmetric group action

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

math.AC↗

arXiv2016

Hilbert Functions of $\mathfrak S_n$-Stable Artinian Gorenstein Algebras

We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.

math.AC↗

arXiv2015

Symmetric tensors: rank and Strassen's conjecture

In this paper we introduce the notion of linear computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

math.AC↗

arXiv2015

$e$-computable forms and the Strassen conjecture

In this paper we introduce the notion of $e$-computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

math.AC↗

arXiv2015

Expressing a General Form as a Sum of Determinants

Let A= (a_{ij}) be a non-negative integer k x k matrix. A is a homogeneous matrix if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. We ask: If A is a homogeneous matrix and if F is a form in C[x_1, \dots x_n] with deg(F) = trace(A), what is the least integer, s(A), so that F = det M_1 + ... + det M_{s(A)}, where the M_i's are k x k matrices of forms with degree matrix A? We consider this problem for n>3 and we prove that s(A) is at most k^{n-3} and s(A) <k^{n-3} in infinitely many cases. However s(A) = k^{n-3} when the entries of A are large with respect to k.

math.AG↗

arXiv2014

The secant line variety to the varieties of reducible plane curves

Let $λ=[d_1,\dots,d_r]$ be a partition of $d$. Consider the variety $\mathbb{X}_{2,λ} \subset \mathbb{P}^N$, $N={d+2 \choose 2}-1$, parameterizing forms $F\in k[x_0,x_1,x_2]_d$ which are the product of $r\geq 2$ forms $F_1,\dots,F_r$, with deg$F_i = d_i$. We study the secant line variety $σ_2(\mathbb{X}_{2,λ})$, and we determine, for all $r$ and $d$, whether or not such a secant variety is defective. Defectivity occurs in infinitely many "unbalanced" cases.

math.AG↗

arXiv2013

Reduced and non-reduced linear spaces: Lines and points

In this paper we consider the problem of determining the Hilbert function of schemes X of the proiective space P^n which are the generic union of s lines and one m-multiple point. We completely solve this problem for any s and m when n > 3. When n=3 we find several defective such schemes and conjecture that they are the only ones. We verify this conjecture in several cases.

math.AG↗

arXiv2012

The Waring rank of the sum of pairwise coprime monomials

In this paper we compute the Waring rank of any polynomial of the form F=M_1+...+M_r, where the M_i are pairwise coprime monomials, i.e., GCD(M_i,M_j)=1 for i not j. In particular, we determine the Waring rank of any monomial. As an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form. As a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials.

math.AC↗

arXiv2011

The Solution to Waring's Problem for Monomials

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve the Waring Problem for all monomials, i.e. we compute the minimal number of linear forms needed to write a monomial as a sum of powers of these linear forms. Moreover, we give an explicit description of a sum of powers decomposition for monomials. We also produce new bounds for the Waring rank of polynomials which are a sum of pairwise coprime monomials.

math.AC↗

arXiv2010

3-dimensional sundials

Robin Hartshorne and Alexander Hirschowitz proved that a generic collection of lines on $\mathbb P^n$, $n \geq 3$, has bipolynomial Hilbert Function. We extended this result to a specialization of the collection of generic lines, by considering a union of lines and $3$-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).

math.AG↗

arXiv2010

Monomials as sums of powers: the Real binary case

We generalize an example, due to Sylvester, and prove that any monomial of degree $d$ in $\mathbb R[x_0, x_1]$, which is not a power of a variable, cannot be written as a linear combination of fewer than $d$ powers of linear forms.

math.AG↗

arXiv2008

The Gotzmann Coefficients of Hilbert Functions

In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric consequences improve some results in this direction first given by Green and extend others by Bigatti, Geramita, and Migliore. Other applications of our detailed investigation of how the Hilbert polynomial is written as a sum of binomials, are to conditions that must be satisfied by a polynomial if it is to be the Hilbert polynomial of a non-degenerate integral subscheme of $\mathbb P^n$ (a problem posed by R. Stanley). We also give some new restrictions on the Hilbert function of a zero dimensional reduced scheme with the Uniform Position Property.

math.AC↗

arXiv1999

Resolutions of Subsets of Finite Sets of Points in Projective Space

Given a finite set, $X$, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is ``as big as possible'' inside $X$. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals of subsets of this finite set? We first give a maximal rank type description of the most generic possible resolution of a subset. Then we show that this generic resolution is not always achieved, by incorporating an example of Eisenbud and Popescu. However, we show that it {\em is} achieved for sets of points in projective two space: given any finite set of points in projective two space for which the minimal free resolution is known, there must exist a subset having the predicted resolution.

math.AG↗