58,622 research outputs found
Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach
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
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
Let be a commutative Noetherian ring, and let be a proper ideal
of , . Consider the polynomial rings and . Suppose that linear equations are
solvable in . It is shown that linear equations are solvable in
(thereby theoretically Gr\"obner bases for ideals of are well defined and
constructible) and that practically Gr\"obner bases in with respect to any
given monomial ordering can be obtained by constructing Gr\"obner bases in ,
and moreover, all basic applications of a Gr\"obner basis at the level of
can be realized by a Gr\"obner basis at the level of . Typical applications
of this result are demonstrated respectively in the cases where is
a PID, is a polynomial ring over a PID , and
is a polynomial ring over a field .Comment: 21page
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