On Descartes’ rule for polynomials with two variations of signs

Cheriha, Hassen; Gati, Yousra; Kostov, Vladimir Petrov

doi:10.1007/s10986-020-09491-9

Cited by 5 publications

(9 citation statements)

References 6 publications

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“…If α 4 = −, then the couple K 4 is realizable and K 4 is not realizable only when K 4 = (Σ 1,3,1 , (0, 2)). This means that σ • = Σ 1,4,1,1,3,1 , with Descartes' pair (5,5). This sign pattern is realizable with the admissible pairs (1, a), a = 3 or 5:…”

Section: Proof Of Theoremmentioning

confidence: 85%

Further than Descartes' rule of signs

Gati¹,

Kostov²,

Tarchi³

2023

Preprint

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The sign pattern defined by the real polynomial Q := Σ d j=0 a j x j , a j = 0, is the string σ(Q) := (sgn(a d ), . . . , sgn(a 0 )). The quantities pos and neg of positive and negative roots of Q satisfy Descartes' rule of signs. A couple (σ 0 , (pos, neg)), where σ 0 is a sign pattern of length d + 1, is realizable if there exists a polynomial Q with pos positive and neg negative simple roots, with (d − pos − neg)/2 complex conjugate pairs and with σ(Q) = σ 0 . We present a series of couples (sign pattern, pair (pos, neg)) depending on two integer parameters and with pos ≥ 1, neg ≥ 1, which is not realizable. For d = 9, we give the exhaustive list of realizable couples with two sign changes in the sign pattern.

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Section: Proof Of Theoremmentioning

confidence: 85%

Further than Descartes' rule of signs

Gati¹,

Kostov²,

Tarchi³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Indeed, for d = 9, there exists a single non-realizable case (modulo the Z 2 × Z 2 -action) with both components of the AP nonzero, this is the couple C := ((+, −, −, −, −, +, +, +, +, −), (1, 6)), see [3]. There exist non-realizable cases in which one of the components of the AP equals 0, see [4]. However there are no non-realizable couples in which the AP equals (1,8), (1,4), (3,6), (8, 1), (4, 1) or (6,3).…”

Section: Properties Of the Discriminant Setmentioning

confidence: 99%

“…There exist non-realizable cases in which one of the components of the AP equals 0, see [4]. However there are no non-realizable couples in which the AP equals (1,8), (1,4), (3,6), (8, 1), (4, 1) or (6,3). For d = 9, there exist non-realizable cases in which one of the components of the AP equals 0, see [5].…”

Section: Properties Of the Discriminant Setmentioning

confidence: 99%

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Degree 5 polynomials and Descartes' rule of signs

Cheriha¹,

Gati²,

Kostov³

2020

Preprint

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For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers pos of positive and neg of negative roots (counted with multiplicity) are majorized respectively by the numbers c and p of sign changes and sign preservartions in the sequence of its coefficients, and that the differences c − pos and p − neg are even numbers. For degree 5 polynomials, it has been proved by A. Albouy and Y. Fu that there exist no such polynomials having three distinct positive and no negative roots and whose signs of the coefficients are (+, +, −, +, −, −) (or having three distinct negative and no positive roots and whose signs of the coefficients are (+, −, −, −, −, +)). For degree 5 and when the leading coefficient is positive, these are all cases of numbers of positive and negative roots (all distinct) and signs of the coefficients which are compatible with Descartes' rule of signs, but for which there exist no such polynomials. We explain this non-existence and the existence in all other cases with d = 5 by means of pictures showing the discriminant set of the family of polynomials x 5 + x 4 + ax 3 + bx 2 + cx + d together with the coordinate axes.

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“…in [22]). Different aspects concerning Descartes' rule of signs are treated in papers [23,5,6,7,8] and [14].…”

Section: Introductionmentioning

confidence: 99%

The disconnectedness of certain sets defined after uni-variate polynomials

Kostov¹

2022

Constructive Mathematical Analysis

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We consider the set of monic real uni-variate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $d\geq 6$ and for signs of the coefficients $(+,-,+,+,\ldots ,+,+,-,+)$, we prove that the set of such polynomials having two positive, $d-4$ negative and two complex conjugate roots, is not connected. For $pos+neg\leq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.

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On Descartes’ rule for polynomials with two variations of signs

Cited by 5 publications

References 6 publications

Further than Descartes' rule of signs

Further than Descartes' rule of signs

Degree 5 polynomials and Descartes' rule of signs

The disconnectedness of certain sets defined after uni-variate polynomials

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