On multivariate Descartes' rule -- a counterexample
Let \(P_1, \dots, P_k\) be real polynomials in \(k\) variables, and \(A_1, \dots, A_k\) be the supports of these polynomials. Equip each \(A_i\) with a distribution \(\sigma_i\) of signs: A point gets the sign (``+'' or ``--'') of the coefficient of the corresponding monomial of \(P_i\). Suppose that the polynomials \(P_1, \dots, P_k\) have a finite number \(r\) of common zeros in \((\mathbb{R}^*)^k\). We can choose for each \(A_i\) a real-valued function \(\omega_i\) defined on \(A_i\), and compare \(r\) with the number of common zeroes in \((\mathbb{R}^*)^k\) of \(T\)-polynomials provided by the combinatorial patchworking [see \textit{I. Itenberg} and \textit{O. Viro}, Math. Intell. 18, No. 4, 19-28 (1996; Zbl 0876.14017) and \textit{B. Sturmfels}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 21, No. 3, 377-386 (1994; Zbl 0826.14032)] with the initial data \((A_1, \sigma_1, \omega_1), \dots, (A_k, \sigma_k, \omega_k)\). The paper is devoted to a counterexample to the following conjecture: (1) There exist functions \(\omega_1, \dots, \omega_k\) such that the number of common zeros in \((\mathbb{R}^*)^k\) of \(T\)-polynomials obtained from the initial data \((A_1,\sigma_1,\omega_1), \dots, (A_k, \sigma_k, \omega_k)\) is at least \(r\). (2) Let \(m\) be an open orthant of \(\mathbb{R}^k\). Then, there exist functions \(\omega_1,\dots,\omega_k\) such that the number of common zeroes in \(m\) of \(T\)-polynomials obtained from the initial data \((A_1,\sigma_1,\omega_1),\dots,(A_k,\sigma_k,\omega_k)\) is greater or equal to the number of common zeros of \(P_1, \dots, P_k\) in \(m\). [In fact, in the paper, the conjecture is formulated in combinatorial terms; for details see \textit{I. Itenberg} and \textit{M.-F. Roy}, Beitr. Algebra Geom. 37, No. 2, 337-346 (1996; Zbl 0870.14038)]. This conjecture is an attempt to generalize Descartes' rule to the case of several variables. The constructed counterexample is a pair of polynomials \(P_1\) and \(P_2\) in two variables contradicting both to (1) and (2).
scientific article; zbMATH DE number 1127634