Mixed Volume Techniques for Embeddings of Laman Graphs

Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is to provide the methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight.Comment: Thorough revision of the first version. (13 pages, 4 figures

Numerical homotopies to compute generic points on positive dimensional algebraic sets

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem

On hardness of computing analytic Brouwer degree

We prove that counting the analytic Brouwer degree of rational coefficient polynomial maps in $\operatorname{Map}(\mathbb C^d, \mathbb C^d)$ -- presented in degree-coefficient form -- is hard for the complexity class $\operatorname{\sharp P}$, in the following sense: if there is a randomized polynomial time algorithm that counts the Brouwer degree correctly for a good fraction of all input instances (with coefficients of bounded height where the bound is an input to the algorithm), then $\operatorname{P}^{\operatorname{\sharp P}} =\operatorname{BPP}$

Elimination for generic sparse polynomial systems

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an l-dimensional coordinate affine space with l < n. The complexity of the algorithm depends polynomially on combinatorial invariants associated to the supports.Comment: 22 page

Multiplicity and Lojasiewicz exponent of generic linear sections of monomial ideals

We obtain a characterisation of the monomial ideals I subset of C[x(1), . . . , x(n)] of finite colength that satisfy the condition e(I) = L-0((1)) (I) . . . L-0((n)) (I), where L-0((1)) (I), . . . , L-0((n)) (I) is the sequence of mixed Lojasiewicz exponents of I and e(I) is the Samuel multiplicity of I. These are the monomial ideals whose integral closure admits a reduction generated by homogeneous polynomials.The author was partially supported by DGICYT Grant MTM2012-33073.Bivià-Ausina, C. (2015). Multiplicity and Lojasiewicz exponent of generic linear sections of monomial ideals. Bulletin of the Australian Mathematical Society. 91(2):191-201. https://doi.org/10.1017/S0004972714001154S191201912Teissier, B. (2012). Some Resonances of Łojasiewicz Inequalities. Wiadomości Matematyczne, 48(2). doi:10.14708/wm.v48i2.337Rojas, J. M., & Wang, X. (1996). Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes. Journal of Complexity, 12(2), 116-133. doi:10.1006/jcom.1996.0009Rees, D. (1988). Lectures on the Asymptotic Theory of Ideals. doi:10.1017/cbo9780511525957Rees, D. (1956). Valuations Associated with a Local Ring (II). Journal of the London Mathematical Society, s1-31(2), 228-235. doi:10.1112/jlms/s1-31.2.228Howald, J. A. (2001). Transactions of the American Mathematical Society, 353(07), 2665-2672. doi:10.1090/s0002-9947-01-02720-9Hickel, M. (2010). Fonction asymptotique de Samuel des sections hyperplanes et multiplicité. Journal of Pure and Applied Algebra, 214(5), 634-645. doi:10.1016/j.jpaa.2009.07.015De Fernex, T., Ein, L., & Mustaţǎ, M. (2004). Multiplicities and log canonical threshold. Journal of Algebraic Geometry, 13(3), 603-615. doi:10.1090/s1056-3911-04-00346-7Bivià-Ausina, C. (2008). Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals. Mathematische Zeitschrift, 262(2), 389-409. doi:10.1007/s00209-008-0380-zBivià-Ausina, C. (2008). Joint reductions of monomial ideals and multiplicity of complex analytic maps. Mathematical Research Letters, 15(2), 389-407. doi:10.4310/mrl.2008.v15.n2.a15Rojas, J. M. (1994). A convex geometric approach to counting the roots of a polynomial system. Theoretical Computer Science, 133(1), 105-140. doi:10.1016/0304-3975(93)00062-a[6] C. Bivià-Ausina and T. Fukui , ‘Mixed Łojasiewicz exponents, log canonical thresholds of ideals and bi-Lipschitz equivalence’, Preprint, 2014, arXiv:1405.2110 [math.AG].Ewald, G. (1996). Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics. doi:10.1007/978-1-4612-4044-0Lejeune-Jalabert, M., & Teissier, B. (2008). Clôture intégrale des idéaux et équisingularité. Annales de la faculté des sciences de Toulouse Mathématiques, 17(4), 781-859. doi:10.5802/afst.1203Bivià-Ausina, C., & Encinas, S. (2012). Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations. Revista Matemática Complutense, 26(2), 773-798. doi:10.1007/s13163-012-0104-0Rees, D. (1984). Generalizations of Reductions and Mixed Multiplicities. Journal of the London Mathematical Society, s2-29(3), 397-414. doi:10.1112/jlms/s2-29.3.397Biviá-Ausina, C. (2004). Nondegenerate Ideals in Formal Power Series Rings. Rocky Mountain Journal of Mathematics, 34(2), 495-511. doi:10.1216/rmjm/1181069864Bivià-Ausina, C. (2005). JACOBIAN IDEALS AND THE NEWTON NON-DEGENERACY CONDITION. Proceedings of the Edinburgh Mathematical Society, 48(1), 21-36. doi:10.1017/s0013091504000173Li, T. Y., & Wang, X. (1996). The BKK root count in $\mathbf{C}^n$. Mathematics of Computation, 65(216), 1477-1485. doi:10.1090/s0025-5718-96-00778-