Arithmetic multivariate Descartes’ rule. (English) Zbl 1046.12001

The author obtains explicit upper bounds for the number of geometrically isolated roots in \({\mathcal L}^n\) of \(F:=(f_1, \dots, f_k) \in {\mathcal L}[x_1, \dots, x_n]\), where \({\mathcal L}\) is any number field for a \({\mathfrak p}\)-adic field. He proves, for example, that \(1+\left({\mathcal C}n(\mu-n)^3\log(\mu-n)\right)^n\) is such a bound, with \({\mathcal C}\) an explicitly computable constant that depends only on \({\mathcal L}\). This result is a multivariate analogue of Descartes’ rule of signs (1637) and of a theorem of A. G. Khovanski on fewnomials [Fewnomials. Translations of Mathematical Monographs, 88. (Providence, RI: American Mathematical Society) (1991; Zbl 0728.12002)]. The proof is obtained through another main theorem about the distribution of \(p\)-adic complex roots close to the point \((1,\dots,1)\) and on a result of A. L. Smirnov [St. Petersbg. Math. J. 8, 651–659 (1997); translation from Algebra Anal. 8, No. 4, 161–172 (1996; Zbl 0883.14030)] on the distribution of norms of \(p\)-adic complex roots. Pertinent applications to additive complexity are obtained as are connections to amoebae. Sharper bounds for particular choices of \(F=(f_1, \dots, f_k)\) are also deduced.


12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
11T06 Polynomials over finite fields
52B11 \(n\)-dimensional polytopes
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