33 research outputs found
A lower bound for the determinantal complexity of a hypersurface
We prove that the determinantal complexity of a hypersurface of degree is bounded below by one more than the codimension of the singular locus,
provided that this codimension is at least . As a result, we obtain that the
determinantal complexity of the permanent is . We also prove
that for , there is no nonsingular hypersurface in of
degree that has an expression as a determinant of a matrix of
linear forms while on the other hand for , a general determinantal
expression is nonsingular. Finally, we answer a question of Ressayre by showing
that the determinantal complexity of the unique (singular) cubic surface
containing a single line is .Comment: 7 pages, 0 figure
Equality of Graver bases and universal Gr\"obner bases of colored partition identities
Associated to any vector configuration A is a toric ideal encoded by vectors
in the kernel of A. Each toric ideal has two special generating sets: the
universal Gr\"obner basis and the Graver basis. While the former is generally a
proper subset of the latter, there are cases for which the two sets coincide.
The most prominent examples among them are toric ideals of unimodular matrices.
Equality of universal Gr\"obner basis and Graver basis is a combinatorial
property of the toric ideal (or, of the defining matrix), providing interesting
information about ideals of higher Lawrence liftings of a matrix. Nonetheless,
a general classification of all matrices for which both sets agree is far from
known. We contribute to this task by identifying all cases with equality within
two families of matrices; namely, those defining rational normal scrolls and
those encoding homogeneous primitive colored partition identities.Comment: minor revision; references added; introduction expanded
Small Chvatal rank
We propose a variant of the Chvatal-Gomory procedure that will produce a
sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax
<= b} as b varies. The number of steps needed is called the small Chvatal rank
(SCR) of A. We characterize matrices for which SCR is zero via the notion of
supernormality which generalizes unimodularity. SCR is studied in the context
of the stable set problem in a graph, and we show that many of the well-known
facet normals of the stable set polytope appear in at most two rounds of our
procedure. Our results reveal a uniform hypercyclic structure behind the
normals of many complicated facet inequalities in the literature for the stable
set polytope. Lower bounds for SCR are derived both in general and for
polytopes in the unit cube.Comment: 24 pages, 3 figures, v3. Major revision: additional author, new
application to stable-set polytopes, reorganization of sections. Accepted for
publication in Mathematical Programmin
An Ehrhart theoretic approach to generalized Golomb rulers
A Golomb ruler is a sequence of integers whose pairwise differences, or
equivalently pairwise sums, are all distinct. This definition has been
generalized in various ways to allow for sums of h integers, or to allow up to
g repetitions of a given sum or difference. Beck, Bogart, and Pham applied the
theory of inside-out polytopes of Beck and Zaslavsky to prove structural
results about the counting functions of Golomb rulers. We extend their approach
to the various types of generalized Golomb rulers.Comment: 15 pages, 2 figure
A parametric version of LLL and some consequences: parametric shortest and closest vector problems
Given a parametric lattice with a basis given by polynomials in Z[t], we give
an algorithm to construct an LLL-reduced basis whose elements are eventually
quasi-polynomial in t: that is, they are given by formulas that are piecewise
polynomial in t (for sufficiently large t), such that each piece is given by a
congruence class modulo a period. As a consequence, we show that there are
parametric solutions of the shortest vector problem (SVP) and closest vector
problem (CVP) that are also eventually quasi-polynomial in t.Comment: 20 pages. Accepted for publication in SIAM Journal on Discrete
Mathematics. Revised title and opening paragraphs, slightly modified
statement of Theorem 1.4, added explanation of some steps in Section 3, and
implemented various minor improvements suggested by anonymous referee