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  • Open Daily: 10am - 10pm
    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
Uni-Variate Polynomials in Analysis: Hyperbolic Polynomials, Order of Moduli, and Descartes' Rule of Signs

Uni-Variate Polynomials in Analysis: Hyperbolic Polynomials, Order of Moduli, and Descartes' Rule of Signs

Kostov, Vladimir P.

Hardcover

Series: de Gruyter Expositions in Mathematics, Book 78

FictionGeneral MathematicsGeometry

ISBN10: 3119143553
ISBN13: 9783119143554
Publisher: de Gruyter
Published: Oct 1 2025
Pages: 112
Weight: 0.84
Height: 0.31 Width: 6.69 Depth: 9.61
Language: English

The book summarizes recent results on problems with uni-variate polynomials. The first of them reads: given the signs of the coefficients of a real polynomial (i. e. its sign pattern), for which pairs of prescribed numbers of positive and negative roots (compatible with Descartes' rule of signs) can one find such a polynomial? For each degree greater or equal to 4, there are non-realizable cases. The problem is resolved for degree less or equal to 8. In another realization problem (resolved for degree less or equal to 5), one fixes the pairs (compatible with Rolle's theorem) of numbers of positive and negative roots of the polynomial and its non-constant derivatives. A third problem concerns polynomials with all roots real. One considers the sign pattern and the order in which the moduli of its positive and negative roots are arranged on the positive half-line. There are examples of pairs (sign pattern, order of moduli) compatible with Descartes' rule of signs that are not realizable. And there are various questions about the discriminant of the general family of uni-variate polynomials. The non-trivial answers to these simply formulated problems will give students and scholars a better understanding of uni-variate polynomials.

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