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Multivariate Sturm-Habicht sequences: real root counting on n-rectangles and triangles.

Laureano González-Vega, Guadalupe Trujillo (1997)

Revista Matemática de la Universidad Complutense de Madrid

The main purpose of this note is to show how Sturm-Habicht Sequence can be generalized to the multivariate case and used to compute the number of real solutions of a polynomial system of equations with a finite number of complex solutions. Using the same techniques, some formulae counting the number of real solutions of such polynomial systems of equations inside n-dimensional rectangles or triangles in the plane are presented.

Niven’s Theorem

Artur Korniłowicz, Adam Naumowicz (2016)

Formalized Mathematics

This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

On Arrangements of Real Roots of a Real Polynomial and Its Derivatives

Kostov, Vladimir (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 12D10.We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its s-th derivative are realized by real polynomials.

On generalized “ham sandwich” theorems

Marek Golasiński (2006)

Archivum Mathematicum

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let A 1 , ... , A m ⊆ ℝ n be subsets with finite Lebesgue measure. Then, for any sequence f 0 , ... , f m of ℝ -linearly independent polynomials in the polynomial ring ℝ [ X 1 , ... , X n ] there are real numbers λ 0 , ... , λ m , not all zero, such that the real affine variety { x ∈ ℝ n ; λ 0 f 0 ( x ) + ⋯ + λ m f m ( x ) = 0 } simultaneously bisects each of subsets A k , k = 1 , ... , m . Then some its applications are studied.

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