58,622 research outputs found

    Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

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    Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure

    Fast Gr\"obner Basis Computation for Boolean Polynomials

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    We introduce the Macaulay2 package BooleanGB, which computes a Gr\"obner basis for Boolean polynomials using a binary representation rather than symbolic. We compare the runtime of several Boolean models from systems in biology and give an application to Sudoku

    On the Construction of Gr\"obner Bases with Coefficients in Quotient Rings

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    Let Ξ›\Lambda be a commutative Noetherian ring, and let II be a proper ideal of Ξ›\Lambda, R=Ξ›/IR=\Lambda /I. Consider the polynomial rings T=Ξ›[x1,...xn]T=\Lambda [x_1,...x_n] and A=R[x1,...,xn]A=R[x_1,...,x_n]. Suppose that linear equations are solvable in Ξ›\Lambda. It is shown that linear equations are solvable in RR (thereby theoretically Gr\"obner bases for ideals of AA are well defined and constructible) and that practically Gr\"obner bases in AA with respect to any given monomial ordering can be obtained by constructing Gr\"obner bases in TT, and moreover, all basic applications of a Gr\"obner basis at the level of AA can be realized by a Gr\"obner basis at the level of TT. Typical applications of this result are demonstrated respectively in the cases where Ξ›=D\Lambda=D is a PID, Ξ›=D[y1,...,ym]\Lambda =D[y_1,...,y_m] is a polynomial ring over a PID DD, and Ξ›=K[y1,...,ym]\Lambda =K[y_1,...,y_m] is a polynomial ring over a field KK.Comment: 21page
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