Arithmetic Multivariate Descartes' Rule Arithmetic Multivariate Descartes' Rule
نویسنده
چکیده
Let L be any number field or p-adic field and consider F := (f1, . . . , fk) where fi∈L[x ±1 1 , . . . , x ±1 n ]\{0} for all i and there are exactlym distinct exponent vectors appearing in f1, . . . , fk. We prove that F has no more than 1+ ( σm(m− 1)2n2 logm )n geometrically isolated roots in Ln, where σ is an explicit and effectively computable constant depending only on L. This gives a significantly sharper arithmetic analogue of Khovanski’s Theorem on Fewnomials and a higher-dimensional generalization of an earlier result of Hendrik W. Lenstra, Jr. for the case of a single univariate polynomial. We also present some further refinements of our new bound.
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