Sparse Gr\"obner Bases: the Unmixed Case

Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a \emph{semigroup algebra}, \emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to solve symbolically sparse systems, we introduce \emph{sparse Gr\"obner bases}, an analog of classical Gr\"obner bases for semigroup algebras, and we propose sparse variants of the $F_5$ and FGLM algorithms to compute them. Our prototype "proof-of-concept" implementation shows large speed-ups (more than 100 for some examples) compared to optimized (classical) Gr\"obner bases software. Moreover, in the case where the generating subset of monomials corresponds to the points with integer coordinates in a normal lattice polytope $\mathcal P\subset\mathbb R^n$ and under regularity assumptions, we prove complexity bounds which depend on the combinatorial properties of $\mathcal P$. These bounds yield new estimates on the complexity of solving $0$-dim systems where all polynomials share the same Newton polytope (\emph{unmixed case}). For instance, we generalize the bound $\min(n_1,n_2)+1$ on the maximal degree in a Gr\"obner basis of a $0$-dim. bilinear system with blocks of variables of sizes $(n_1,n_2)$ to the multilinear case: $\sum n_i - \max(n_i)+1$. We also propose a variant of Fr\"oberg's conjecture which allows us to estimate the complexity of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan (2014

Quillen homology for operads via Gr\"obner bases

The main goal of this paper is to present a way to compute Quillen homology of operads. The key idea is to use the notion of a shuffle operad we introduced earlier; this allows to compute, for a symmetric operad, the homology classes and the shape of the differential in its minimal model, although does not give an insight on the symmetric groups action on the homology. Our approach goes in several steps. First, we regard our symmetric operad as a shuffle operad, which allows to compute its Gr\"obner basis. Next, we define a combinatorial resolution for the "monomial replacement" of each shuffle operad (provided by the Gr\"obner bases theory). Finally, we explain how to "deform" the differential to handle every operad with a Gr\"obner basis, and find explicit representatives of Quillen homology classes for a large class of operads. We also present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin-Vilkovisky algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint arXiv:0912.4895. Final version, to appear in Documenta Mat

Numerical Schubert calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

Towards Mixed Gr{\"o}bner Basis Algorithms: the Multihomogeneous and Sparse Case

One of the biggest open problems in computational algebra is the design of efficient algorithms for Gr{\"o}bner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed polynomial systems, that is systems with polynomials having the same support, using the approach of Faug{\`e}re, Spaenlehauer, and Svartz [ISSAC'14]. We present two algorithms for sparse Gr{\"o}bner bases computations for mixed systems. The first one computes with mixed sparse systems and exploits the supports of the polynomials. Under regularity assumptions, it performs no reductions to zero. For mixed, square, and 0-dimensional multihomogeneous polynomial systems, we present a dedicated, and potentially more efficient, algorithm that exploits different algebraic properties that performs no reduction to zero. We give an explicit bound for the maximal degree appearing in the computations

Uniform bounds on multigraded regularity

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann's regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes.Comment: 23 pages, 2 figure

Quivers from Matrix Factorizations

We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single CP1 but have "length" greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau-Ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions.Comment: 33 pages, (minor changes