Details:
Title  Gröbner bases of bihomogeneous ideals generated by polynomials of bidegree : Algorithms and complexity  Author(s)  JeanCharles Faugère, Mohab Safey, PierreJean Spaenlehauer  Type  Article in Journal  Abstract  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 for 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 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. 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 0dimensional 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(((n_x+n_y+min(n_x+1,n_y+1) ((min(nx+1,n_y+1))^ω) , which is polynomial in n_x+n_y (i.e. the number of unknowns) when min(n_x,n_y) is constant.  Keywords  Gröbner bases, Bihomogeneous ideals, Algorithms, Complexity  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717110001902 
Language  English  Journal  Journal of Symbolic Computation  Volume  46  Number  4  Pages  406  437  Year  2011  Edition  0  Translation 
No  Refereed 
No 
