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We consider real univariate polynomials with all roots real. Such a polynomial with c sign changes and p sign preservations in the sequence of its coefficients has c positive and p negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is. [ABSTRACT FROM AUTHOR]

By Descartes' rule of signs, a real degree d polynomial P with all nonvanishing coefficients with c sign changes and p sign preservations in the sequence of its coefficients (c + p = d) has pos 6 c positive and neg ≡ p negative roots, where pos ≡ c (mod 2) and neg ≤ p (mod 2). For 1 ≤ d ≤ 3, for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair (pos, neg) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly neg negative roots (all of them simple). For d ≥ 4 this is not so. It was observed that for 4 ≤ d ≤ 8, in all nonrealizable cases either pos = 0 or neg = 0. It was conjectured that this is the case for any d ≥ 4. We show a counterexample to this conjecture for d = 11. Namely, we prove that for the sign pattern (+, -, -, -, -, -, +, +, +, +, +, -) and the pair (1, 8) there exists no polynomial with 1 positive, 8 negative simple roots and a complex conjugate pair. [ABSTRACT FROM AUTHOR]

We give a Descartes'-like bound on the number of positive solutions to a system of fewnomials that holds when its exponent vectors are not in convex position and a sign condition is satisfied. This was discovered while developing algorithms and software for computing the Gale transform of a fewnomial system, which is our main goal. This software is a component of a package we are developing for Khovanskii-Rolle continuation, which is a numerical algorithm to compute the real solutions to a system of fewnomials. [ABSTRACT FROM AUTHOR]

Given convex polytopes $$P_1,\ldots ,P_r \subset \mathbb {R}^n$$ and finite subsets $$\mathcal {W}_I$$ of the Minkowski sums $$P_I=\sum _{i \in I} P_i$$ , we consider the quantity $$N(\mathbf {W})=\sum _{I \subset \mathbf{[}r \mathbf{]} } {(-1)}^{r-|I|} \big | \mathcal {W}_I \big |$$ . If $$\mathcal {W}_I=\mathbb {Z}^n \cap P_I$$ and $$P_1,\ldots ,P_n$$ are lattice polytopes in $$\mathbb {R}^n$$ , then $$N(\mathbf {W})$$ is the classical mixed volume of $$P_1,\ldots ,P_n$$ giving the number of complex solutions of a general complex polynomial system with Newton polytopes $$P_1,\ldots ,P_n$$ . We develop a technique that we call irrational mixed decomposition which allows us to estimate $$N(\mathbf {W})$$ under some assumptions on the family $$\mathbf {W}=(\mathcal {W}_I)$$ . In particular, we are able to show the nonnegativity of $$N(\mathbf {W})$$ in some important cases. A special attention is paid to the family $$\mathbf {W}=(\mathcal {W}_I)$$ defined by $$\mathcal {W}_I=\sum _{i \in I} \mathcal {W}_i$$ , where $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ are finite subsets of $$P_1,\ldots ,P_r$$ . The associated quantity $$N(\mathbf {W})$$ is called discrete mixed volume of $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ . Using our irrational mixed decomposition technique, we show that for $$r=n$$ the discrete mixed volume is an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $$\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$$ . We also prove that the discrete mixed volume associated with $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ is bounded from above by the Kouchnirenko number $$\prod _{i=1}^r (|\mathcal {W}_i|-1)$$ . For $$r=n$$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $$\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$$ . This conjecture was disproved, but our result show that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking. [ABSTRACT FROM AUTHOR]

The article discusses constant power line sequences and analyzes relative geometric position of the roots as coefficients are allowed to depend on n. Topics include application of Descartes's rule of signs, use of squeeze theorem and continuity of the polynomials in case of zero, and a theorem assuming two sequences of positive real numbers in case of odd numbers.

The article offers information on the use of Descartes' rule of signs and certain properties of polynomial functions to estimate the maximum number of turning points. It states that the number of turning points in a graph is connected to both the oscillatory and complexing behavior of the function. The important focus of the intermediate value theorem, Descartes' rule of signs, and the factor theorem on the roots of polynomial equations is cited.