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Solving multihomogeneous systems, as a wide range of structured algebraicsystems occurring frequently in practical problems, is of first importance.Experimentally, solving these systems with Gr\"obner bases algorithms seems tobe easier than solving homogeneous systems of the same degree. Nevertheless,the reasons of this behaviour are not clear. In this paper, we focus onbilinear systems (i.e. bihomogeneous systems where all equations have bidegree(1,1)). Our goal is to provide a theoretical explanation of the aforementionnedexperimental behaviour and to propose new techniques to speed up the Gr\"obnerbasis computations by using the multihomogeneous structure of those systems.The contributions are theoretical and practical. First, we adapt the classicalF5 criterion to avoid reductions to zero which occur when the input is a set ofbilinear polynomials. We also prove an explicit form of the Hilbert series ofbihomogeneous ideals generated by generic bilinear polynomials and give a newupper bound on the degree of regularity of generic affine bilinear systems.This leads to new complexity bounds for solving bilinear systems. We proposealso a variant of the F5 Algorithm dedicated to multihomogeneous systems whichexploits a structural property of the Macaulay matrix which occurs on suchinputs. Experimental results show that this variant requires less time andmemory than the classical homogeneous F5 Algorithm.

Gr\"obner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity