Multihomogeneous Systems Research Articles

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

Gröbner bases of bihomogeneous ideals generated by polynomials of bidegree [formula omitted]: 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 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 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 ( n x + n y + min ( n x + 1 , n y + 1 ) min ( n x + 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.

The number of solutions to some systems of equations

An estimate for the number of solutions to multi-homogeneous systems of equations is obtained by using properties of the Hilbert function of multi-homogeneous ideals.

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N → ∞ , the expected number of Nash equilibria is approximately ( π log N / 2 ) M − 1 / M . Letting M = N → ∞ , the expected number of Nash equilibria is exp ( N M + O ( log N ) ) , where M ≈ 0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.

Multihomogeneous resultant formulae by means of complexes

The first step in the generalization of the classical theory of homogeneous equations to the case of arbitrary support is to consider algebraic systems with multihomogeneous structure. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, and including matrices of hybrid type of these two. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman (J. Algebra, 163 (1994) 115; J. Algebraic Geom., 3 (1994) 569). One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. Whenever such formulae exist, we specify the underlying complexes so as to make the resultant matrix explicit. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bézout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with examples showing the kinds of matrices that may be encountered, and illustrations of our Maple implementation.

The expected number of real roots of a multihomogeneous system of polynomial equations

The mean number of roots of a multihomogeneous system of polynomial equations (with respect to a natural distribution on the space of coefficient vectors) is shown to be at least as large as the square root of the generic number of complex roots, as determined by Bernshtein's theorem. We first extend the methods of Shub and Smale to the class of multihomogeneous systems of polynomial equations, yielding Theorem A, which is a formula expressing the mean number of real roots as a multiple of the mean absolute value of the determinant of a random matrix. Theorem B derives closed form expressions for the mean in special cases that include: (a) Shub and Smale's result that the expected number of real roots of the general homogeneous system is the square root of the generic number of complex roots given by Bezout's theorem; (b) Rojas' characterization of the mean number of real roots of an "unmixed" multihomogeneous system. Special properties of normal random variables are exploited to derive Theorem C, which gives recursive inequalities satisfied by the mean, and the bounds on the mean number of real roots (we also give an upper bound) are derived by comparing these with recursive equalities satisfied by the numbers given by Bernshtein's theorem.