59 research outputs found
The symmetric signature
We define two related invariants for a -dimensional local ring
called syzygy and differential symmetric signature by
looking at the maximal free splitting of reflexive symmetric powers of two
modules: the top dimensional syzygy module of the residue
field and the module of K\"ahler differentials of over .
We compute these invariants for two-dimensional ADE singularities obtaining
, where is the order of the acting group, and for cones over
elliptic curves obtaining for the differential symmetric signature. These
values coincide with the F-signature of such rings in positive characteristic.Comment: Shortened the proofs of Proposition 2.8 and Theorem 3.15; modified
Lemma 3.11; added Remark 3.6, Lemma 4.10, and Lemma 4.11; minor typos fixed;
improved exposition; updated reference
The complexity of MinRank
In this note, we leverage some of our results from arXiv:1706.06319 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 Faug\`ere, Safey El Din, Spaenlehauer
(arXiv:1112.4411).Comment: Corrected a typo in the formula of the main theore
Solving multivariate polynomial systems and an invariant from commutative algebra
The complexity of computing the solutions of a system of multivariate
polynomial equations by means of Gr\"obner bases computations is upper bounded
by a function of the solving degree. In this paper, we discuss how to
rigorously estimate the solving degree of a system, focusing on systems arising
within public-key cryptography. In particular, we show that it is upper bounded
by, and often equal to, the Castelnuovo Mumford regularity of the ideal
generated by the homogenization of the equations of the system, or by the
equations themselves in case they are homogeneous. We discuss the underlying
commutative algebra and clarify under which assumptions the commonly used
results hold. In particular, we discuss the assumption of being in generic
coordinates (often required for bounds obtained following this type of
approach) and prove that systems that contain the field equations or their fake
Weil descent are in generic coordinates. We also compare the notion of solving
degree with that of degree of regularity, which is commonly used in the
literature. We complement the paper with some examples of bounds obtained
following the strategy that we describe
The symmetric signature
This is the author's Ph.D. thesis. We introduce two related invariants for
local (and standard graded) rings called differential and syzygy symmetric
signature. These are defined by looking at the maximal free splitting of the
module of K\"ahler differentials and of the the top-dimensional syzygy module
of the residue field respectively. We study and compute them for different
classes of rings: two-dimensional ADE singularities, two-dimensional cyclic
singularities, and cones over plane elliptic curves (for the differential
symmetric signature). The values obtained coincide with the F-signature of such
rings in positive characteristic. The thesis contains also a short survey on
the Auslander correspondence for quotient singularities.Comment: The main results of the thesis appeared also in arXiv:1602.07184 and
arXiv:1603.0642
Cohomological dimension and arithmetical rank of some determinantal ideals
Let be a non-generic matrix of linear forms in a
polynomial ring. For large classes of such matrices, we compute the
cohomological dimension (cd) and the arithmetical rank (ara) of the ideal
generated by the -minors of . Over an algebraically closed
field, any -matrix of linear forms can be written in the
Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and
nilpotent blocks. B\u{a}descu and Valla computed when
is a concatenation of scroll blocks. In this case we compute
and extend these results to concatenations of Jordan
blocks. Eventually we compute and
in an interesting mixed case, when contains both Jordan and scroll blocks.
In all cases we show that is less than the arithmetical
rank of the determinantal ideal of a generic matrix
F-signature function of quotient singularities
We study the shape of the F-signature function of a d-dimensional quotient singularity k\u301ax1,\u2026,xd\u301bG, 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 86Gl(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
A Pascal's theorem for rational normal curves
Pascal's Theorem gives a synthetic geometric condition for six points
in to lie on a conic. Namely, that the intersection
points , ,
are aligned. One could ask an analogous
question in higher dimension: is there a coordinate-free condition for
points in to lie on a degree 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 ordered points
in 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.Comment: 16 pages, 1 figure. Comments are welcom
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