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

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.Comment: 17 page

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.Comment: 12 pages; Section 5 added in this versio

Fat Points, Inverse Systems, and Piecewise Polynomial Functions

AbstractWe explore the connection between ideals of fat points (which correspond to subschemes of Pnobtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on ad-dimensional simplicial complex Δ embedded inRd. Using the inverse system approach introduced by Macaulay [11], we give a complete characterization of the free resolutions possible for ideals ink[x,y] generated by powers of homogeneous linear forms (we allow the powers to differ). We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed Δ, the dimension of the vector space of splines on Δ of degree less than or equal tok. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree

Complete intersections on general hypersurfaces

We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in \PP^n. We give a complete answer to this question when $2r \leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection

The Gotzmann coefficients of Hilbert functions

AbstractIn 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 Pn (a problem posed by R.P. Stanley). We also give some new restrictions on the Hilbert function of a zero-dimensional reduced scheme with the Uniform Position Property