Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems

B{\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields. When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically. We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all. For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency. However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\le$ \sqrt 1 + 8{\nu} -- 1 holds for a quadratic fewnomial system -- where {\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K). Moreover this certificate can be computed within a polynomial number of arithmetic operations. Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares. More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\Omega$(n 1/2+$\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows. Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201

Lifting Grobner bases from the exterior algebra

In the article "Non-commutative Grobner bases for commutative algebras", Eisenbud-Peeva-Sturmfels proved a number of results regarding Grobner bases and initial ideals of those ideals in the free associative algebra which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one notion of generic initial ideals in the free assoicative algebra, and show that gin's of ideals containing the commutator ideal, or the anti-commutator ideal, are finitely generated.Comment: 6 pages, LaTeX2

Multigraded Castelnuovo-Mumford Regularity

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.Comment: 30 pages, 5 figure

Syzygies of torsion bundles and the geometry of the level l modular variety over M_g

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian. These statements can be viewed an analogues of Green's Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_{g,l} of twisted level l curves of genus g and use this to derive results about the birational geometry of R_{g, l}. For instance, we prove that R_{g,3} is a variety of general type when g>11 and the Kodaira dimension of R_{11,3} is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.Comment: 35 pages, appeared in Invent Math. We correct an inaccuracy in the statement of Prop 2.

Binomial Ideals and Congruences on Nn

Producción CientíficaA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).National Science Foundation (grant DMS-1500832)Ministerio de Economía, Industria y Competitividad (project MTM2015-65764-C3-1)Junta de Extremadura (grupo de investigación FQM-024

Probing scattering phase shifts by attosecond streaking

Attosecond streaking is one of the most fundamental processes in attosecond science allowing for a mapping of temporal (i.e. phase) information on the energy domain. We show that on the single-particle level attosecond streaking time shifts contain spectral phase information associated with the Eisenbud-Wigner-Smith (EWS) time delay, provided the influence of the streaking infrared field is properly accounted for. While the streaking phase shifts for short-ranged potentials agree with the associated EWS delays, Coulomb potentials require special care. We show that the interaction between the outgoing electron and the combined Coulomb and IR laser fields lead to a streaking phase shift that can be described classically

GG-prime and GG-primary GG-ideals on GG-schemes

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties.Comment: 54pages, added Example 6.16 and the reference [8]. The final versio

Possible eta-mesic 3He states within the finite rank approximation

We extend the method of time delay proposed by Eisenbud and Wigner, to search for unstable states formed by eta mesons and the 3He nucleus. Using few body equations to describe eta-3He elastic scattering, we predict resonances and unstable bound states within different models of the eta-N interaction. The eta-3He states predicted within this novel approach are in agreement with the recent claim of the evidence of eta-mesic 3He made by the TAPS collaboration.Comment: 10 pages LaTex, 3 figure

Exceptional collections and D-branes probing toric singularities

We demonstrate that a strongly exceptional collection on a singular toric surface can be used to derive the gauge theory on a stack of D3-branes probing the Calabi-Yau singularity caused by the surface shrinking to zero size. A strongly exceptional collection, i.e., an ordered set of sheaves satisfying special mapping properties, gives a convenient basis of D-branes. We find such collections and analyze the gauge theories for weighted projective spaces, and many of the Y^{p,q} and L^{p,q,r} spaces. In particular, we prove the strong exceptionality for all p in the Y^{p,p-1} case, and similarly for the Y^{p,p-2r} case.Comment: 49 pages, 6 figures; v2 refs added; v3 published versio

Twisted K-Theory of Lie Groups

I determine the twisted K-theory of all compact simply connected simple Lie groups. The computation reduces via the Freed-Hopkins-Teleman theorem to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al.Comment: 16 page