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Gr\"obner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity
Solving multihomogeneous systems, as a wide range of structured algebraic
systems occurring frequently in practical problems, is of first importance.
Experimentally, solving these systems with Gr\"obner bases algorithms seems to
be easier than solving homogeneous systems of the same degree. Nevertheless,
the reasons of this behaviour are not clear. In this paper, we focus on
bilinear systems (i.e. bihomogeneous systems where all equations have bidegree
(1,1)). Our goal is to provide a theoretical explanation of the aforementionned
experimental behaviour and to propose new techniques to speed up the Gr\"obner
basis computations by using the multihomogeneous structure of those systems.
The contributions are theoretical and practical. First, we adapt the classical
F5 criterion to avoid reductions to zero which occur when the input is a set of
bilinear polynomials. We also prove an explicit form of the Hilbert series of
bihomogeneous ideals generated by generic bilinear polynomials and give a new
upper bound on the degree of regularity of generic affine bilinear systems.
This leads to new complexity bounds for solving bilinear systems. We propose
also a variant of the F5 Algorithm dedicated to multihomogeneous systems which
exploits a structural property of the Macaulay matrix which occurs on such
inputs. Experimental results show that this variant requires less time and
memory than the classical homogeneous F5 Algorithm.Comment: 31 page
Trees, parking functions, syzygies, and deformations of monomial ideals
For a graph G, we construct two algebras, whose dimensions are both equal to
the number of spanning trees of G. One of these algebras is the quotient of the
polynomial ring modulo certain monomial ideal, while the other is the quotient
of the polynomial ring modulo certain powers of linear forms. We describe the
set of monomials that forms a linear basis in each of these two algebras. The
basis elements correspond to G-parking functions that naturally came up in the
abelian sandpile model. These ideals are instances of the general class of
monotone monomial ideals and their deformations. We show that the Hilbert
series of a monotone monomial ideal is always bounded by the Hilbert series of
its deformation. Then we define an even more general class of monomial ideals
associated with posets and construct free resolutions for these ideals. In some
cases these resolutions coincide with Scarf resolutions. We prove several
formulas for Hilbert series of monotone monomial ideals and investigate when
they are equal to Hilbert series of deformations. In the appendix we discuss
the sandpile model.Comment: 33 pages; v2: appendix on sandpiles added, references added, typos
corrected; v3: references adde
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