732 research outputs found
F-signature function of quotient singularities
No full textWe study the shape of the F-signature function of a d-dimensional quotient singularity k〚x1,...,xd〛G, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group G⊆Gl(d,k). When G is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples
The Complexity of MinRank
No full textIn this note, we leverage some of our previous results to produce a concise and rigorous proof for the complexity of the generalized MinRank Problem in the under-defined and well-defined case. Our main theorem recovers and extends previous results by Faugère, Safey El Din, Spaenlehauer
The complexity of solving Weil restriction systems
No full textThe solving degree of a system of multivariate polynomial equations provides
an upper bound for the complexity of computing the solutions of the system via
Groebner bases methods. In this paper, we consider polynomial systems that are
obtained via Weil restriction of scalars. The latter is an arithmetic
construction which, given a finite Galois field extension ,
associates to a system defined over a system
defined over , in such a way that the solutions
of over and those of over
are in natural bijection. In this paper, we find upper bounds for the
complexity of solving a polynomial system obtained
via Weil restriction in terms of algebraic invariants of the system
.Comment: Final version. To appear in Journal of Algebr
Quadratic Modelings of Syndrome Decoding
No full textThis paper presents enhanced reductions of the bounded-weight and exact-weight Syndrome Decoding Problem (SDP) to a system of quadratic equations. Over F2, we improve on a previous work and study the degree of regularity of the modeling of the exact weight SDP. Additionally, we introduce a novel technique that transforms SDP instances over Fq into systems of polynomial equations and thoroughly investigate the dimension of their varieties. Experimental results are provided to evaluate the complexity of solving SDP instances using our models through Gröbner bases techniques
Determinantal Varieties From Point Configurations on Hypersurfaces
No full textWe consider the scheme parametrizing ordered points in projective space that lie on a common hypersurface of degree . We show that this scheme has a determinantal structure and we prove that it is irreducible, Cohen--Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of in terms of Castelnuovo--Mumford regularity and -normality. This yields a characterization of the singular locus of and
Simplices Osculating Rational Normal Curves
No full textA classical result of von Staudt states that if eight planes osculate a twisted cubic curve and we divide them into two groups of four, then the eight vertices of the corresponding tetrahedra lie on a twisted cubic curve. In the current paper, we give an alternative proof of this result using modern tools, and at the same time we prove the analogous result for rational normal curves in any projective space. This higher dimensional generalization was claimed without proof in a paper of H.S. White in 1921
Solving degree, last fall degree, and related invariants
No full textIn this paper we study and relate several invariants connected to the solving
degree of a polynomial system. This provides a rigorous framework for
estimating the complexity of solving a system of polynomial equations via
Groebner bases methods. Our main results include a connection between the
solving degree and the last fall degree and one between the degree of
regularity and the Castelnuovo-Mumford regularity.Comment: Final version. To appear in Journal of Symbolic Computatio
Point configurations, phylogenetic trees, and dissimilarity vectors
No full textIn 2004 Pachter and Speyer introduced the higher dissimilarity maps for
phylogenetic trees and asked two important questions about their relation to
the tropical Grassmannian. Multiple authors, using independent methods,
answered affirmatively the first of these questions, showing that dissimilarity
vectors lie on the tropical Grassmannian, but the second question, whether the
set of dissimilarity vectors forms a tropical subvariety, remained opened. We
resolve this question by showing that the tropical balancing condition fails.
However, by replacing the definition of the dissimilarity map with a weighted
variant, we show that weighted dissimilarity vectors form a tropical subvariety
of the tropical Grassmannian in exactly the way that Pachter--Speyer
envisioned. Moreover, we provide a geometric interpretation in terms of
configurations of points on rational normal curves and construct a finite
tropical basis that yields an explicit characterization of weighted
dissimilarity vectors.Comment: Final version. To appear in Proceedings of the National Academy of
Sciences of the United States of America (PNAS
The new generation magnetic-iron-detector to measure the iron overload in the human liver
No full textA Pascal's theorem for rational normal curves
No full textPascal's theorem gives a synthetic geometric condition for six points a,...,f in P2 to lie on a conic. Namely, that the intersection points ab intersect de, af intersect dc, ef intersect bc are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d+4 points in Pd to lie on a degree d rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of d+4-ordered points in Pd that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic
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