Arithmetic multivariate Descartes' rule

<abstract abstract-type="TeX"><p>Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be any number field or p-adic field and consider <i>F</i>:=(<i>f</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>, . . . , <i>f<sub>k</sub></i>) where <i>f</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>, . . . , <i>f<sub>k</sub></i> ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /][<i>x</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>, . . . , <i>x<sub>n</sub></i>] and no more than μ distinct exponent vectors occur in the monomial term expansions of the <i>f<sub>i</sub></i>. We prove that <i>F</i> has no more than 1 + (<i>C<sub>n</sub></i>(μ - <i>n</i>)<sup>3</sup> log (μ - <i>n</i>))<sup><i>n</i></sup> geometrically isolated roots in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]<sup><i>n</i></sup>, where <i>C</i> is an explicit and effectively computable constant depending only on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]. This gives a significantly sharper arithmetic analogue of Khovanski's Theorem on Real Fewnomials and a higher-dimensional generalization of an earlier result of Hendrik W. Lenstra, Jr. for the special case of a single univariate polynomial. We also present some further refinements of our new bounds and a higher-dimensional generalization of a bound of Lipshitz on <i>p</i>-adic complex roots. Connections to non-Archimedean amoebae and computational complexity (including additive complexity and solving for the geometrically isolated rational roots) are discussed along the way. We thus provide the foundations for an effective arithmetic analogue of fewnomial theory.

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