Van der Waerden Conjecture and Applications

Егорычев, Г. П.

doi:10.1016/s1570-7954(96)80005-2

Cited by 3 publications

(2 citation statements)

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“…Our analysis of this quantity employs the following concept. The permanent (e.g., [Ego96]) of an m × n matrix D with entries…”

Section: The Bkk Bound For Multihomogeneous Systemsmentioning

confidence: 99%

“…Below (see also [McL98]) we describe how these formulas can be seen as the consequence of expressing the Bernshtein number for such a system as the permanent (e.g. [Ego96]) of a matrix, after which the recursions are obtained by expanding along a row or column. In investigating whether the mean number of real roots is greater than the square root of the Bernshtein number, as asserted by Theorem 3, it is natural to guess that the squares of the mean numbers of real roots obey the corresponding recursive inequalities, which is the assertion of Theorem 4, since then Theorem 3 follows from induction.…”

Section: Introductionmentioning

confidence: 99%

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

McLennan¹

2002

ajm

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Abstract. 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.

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“…Our analysis of this quantity employs the following concept. The permanent (e.g., [Ego96]) of an m × n matrix D with entries…”

Section: The Bkk Bound For Multihomogeneous Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%