## Section: New Results

### Solving structured systems

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 Gröbner
bases algorithms seems to be easier than solving homogeneous systems
of the same degree. Nevertheless, the reasons of this behaviour are
not clear. In [16] , 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
aforementioned experimental behaviour and to propose new techniques
to speed up the Gröbner basis computations by using the
multihomogeneous structure of those systems. The contributions are
theoretical and practical. First, we adapt the classical ${F}_{5}$
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 ${F}_{5}$
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 ${F}_{5}$ Algorithm.
Lastly, we investigate the complexity of computing a Gröbner
basis for the grevlex ordering of a generic 0-dimensional affine
bilinear system over $k[{x}_{1},...,{x}_{{n}_{x}},{y}_{1},...,{y}_{{n}_{y}}]$. In
particular, we show that this complexity is upper bounded by
$O\left({\left(\begin{array}{c}{n}_{x}+{n}_{y}+min({n}_{x}+1,{n}_{y}+1)\\ min({n}_{x}+1,{n}_{y}+1)\end{array}\right)}^{\omega}\right)$,
which is polynomial in ${n}_{x}+{n}_{y}$ (i.e. the number of unknowns) when
$min({n}_{x},{n}_{y})$ is constant.