The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations
نویسندگان
چکیده
The methods of Shub and Smale [SS93] are extended to the class of multihomogeneous systems of polynomial equations, yielding Theorem 1, which is a formula expressing the mean (with respect to a particular distribution on the space of coefficient vectors) number of real roots as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 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’ [Roj96] characterization of the mean number of real roots of an “unmixed” multihomogeneous system. Theorem 3 gives upper and lower bounds for the mean number of roots, where the lower bound is the square root of the generic number of complex roots, as determined by Bernstein’s [Ber75] theorem. These bounds are derived by induction from recursive inequalities given in Theorem 4. Research at MSRI is supported in part by NSF grant DMS-9701755. I have benefited from numerous discussions with Maurice Rojas, and I am grateful for comments by seminar participants at the University of Minnesota, the AMS Meeting at Temple University in April 1998, and the conference on polynomial system solving at the Mathematical Sciences Research Institute in September 1998. 2 ANDREW MCLENNAN The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations
منابع مشابه
On Classifications of Random Polynomials
Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...
متن کاملCanonical representation for approximating solution of fuzzy polynomial equations
In this paper, the concept of canonical representation is proposed to find fuzzy roots of fuzzy polynomial equations. We transform fuzzy polynomial equations to system of crisp polynomial equations, this transformation is perform by using canonical representation based on three parameters Value, Ambiguity and Fuzziness.
متن کاملCoupled systems of equations with entire and polynomial functions
We consider the coupled system$F(x,y)=G(x,y)=0$,where$$F(x, y)=bs 0 {m_1} A_k(y)x^{m_1-k}mbox{ and } G(x, y)=bs 0 {m_2} B_k(y)x^{m_2-k}$$with entire functions $A_k(y), B_k(y)$.We derive a priory estimates for the sums of the rootsof the considered system andfor the counting function of roots.
متن کاملOn the Expected Number of Real Roots of a System of Random Polynomial Equations
We unify and generalize several known results about systems of random polynomials. We first classify all orthogonally invariant normal measures for spaces of polynomial mappings. For each such measure we calculate the expected number of real zeros. The results for invariant measures extend to underdetermined systems, giving the expected volume for orthogonally invariant random real projective v...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998