Could René Descartes have known this?

Forsgård, Jens; Kostov, Vladimir Petrov; Shapiro, Boris

doi:10.48550/arxiv.1501.00856

Cited by 2 publications

(22 citation statements)

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Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 77%

“…The present paper finishes the study which was begun in [3] of sign patterns and their admissible pairs for polynomials of degree up to 8. The present introduction reproduces with some small modifications the one of [3] and the results obtained in that paper, see Theorems 1, 2 and 3. The new results are given in Theorem 4 and then presented in another way (suitable to be compared to the previously obtained ones) at the end of this section.…”

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 77%

“…In Theorems 1, 2 and 3 we change at some places (w.r.t. the original formulations in [1] or [3]) a sign pattern σ to σ m and the corresponding pair (p, n) to (n, p) in order to have mostly pairs of the form (0, n) in the formulations: Theorem 1. (1) For degree 1, 2 and 3, any sign pattern is realizable with any of its admissible pairs.…”

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 99%

“…The cases d = 7 and d = 8 have been considered in [3]. The exhaustive answer to the question of realizability for d = 7 is as follows: For d = 8 the partial answer from [3] can be summarized by the following theorem. In [3] this result is formulated differently, but equivalently.…”

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 99%

“…The exhaustive answer to the question of realizability for d = 7 is as follows: For d = 8 the partial answer from [3] can be summarized by the following theorem. In [3] this result is formulated differently, but equivalently. In particular, the authors of [3] have not noticed that the number of cases for which the answer still remained unknown can be decreased by one due to the standard Z 2 ×Z 2 -action.…”

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 99%

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On realizability of sign patterns by real polynomials

Kostov¹

2018

Czech.Math.J.

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The classical Descartes' rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers (p, n), chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.

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Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 77%

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 77%

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 99%

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 99%

Section: Formulation Of the Problem And Of The Resultsmentioning

confidence: 99%

See 3 more Smart Citations

On realizability of sign patterns by real polynomials

Kostov¹

2018

Czech.Math.J.

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Polynomials, sign patterns and Descartes' rule of signs

Kostov¹

2018

Math.Boh.

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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 ≤ 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 ≤ 10, 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.

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Could René Descartes have known this?

Cited by 2 publications

References 0 publications

On realizability of sign patterns by real polynomials

On realizability of sign patterns by real polynomials

Polynomials, sign patterns and Descartes' rule of signs

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