The disconnectedness of certain sets defined after uni-variate polynomials

Kostov, Vladimir Petrov

doi:10.33205/cma.1111247

Cited by 3 publications

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“…For its tropical analog see [6]. A related problem concerning polynomials in one variable is considered in [15]. A degree d real polynomial Q := d j=0 a j x j is hyperbolic if all its roots are real.…”

Section: Introductionmentioning

confidence: 99%

Beyond Descartes’ rule of signs

Kostov¹

2023

Constructive Mathematical Analysis

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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.

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Section: Introductionmentioning

confidence: 99%

Beyond Descartes’ rule of signs

Kostov¹

2023

Constructive Mathematical Analysis

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“…[15], Theorem 1). (1) For d ≥ 6, the set P d σ•,(2,d−4) is non-empty and consists of more than one component.…”

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confidence: 99%

“…Notice that some of P d σ,(ℓ + ,ℓ − ) might be empty, see examples below. The problem of non-emptiness for the sets P d σ,(ℓ + ,ℓ − ) has been discussed in some detail in [8,15,16] and a number of follow-up papers. The problem of disconnectedness of P d σ,(ℓ + ,ℓ − ) has been mentioned in [15,Problem 2], where some special cases have been treated.…”

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Real univariate polynomials with given signs of coefficients and simple real roots

Kostov

2024

Mat. Stud.

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We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\ell ^+$ and $\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $\mathcal{P}^d:=\{ P:=x^d+a_1x^{d-1}+\dots +a_d\}$, $a_j\in \mathbb{R}^*=\mathbb{R}\setminus \{ 0\}$, the corresponding sign patterns $\sigma=(\sigma_1,\sigma_2,\dots, \sigma_d)$, where $\sigma_j=$sign$(a_j)$, and the sets $\mathcal{P}^d_{\sigma ,(\ell ^+,\ell ^-)}\subset \mathcal{P}^d$ of polynomials with given triples $(\sigma ,(\ell ^+,\ell ^-))$.We prove that for degree $d\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $\sigma =(-,-,-,+)$, $\ell^+=0$, $\ell ^-=2$, with $d=5$, $\sigma =(-,-,-,-,+)$, $\ell^+=0$, $\ell ^-=3$, and the ones obtained from them under the $\mathbb{Z}_2\times \mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting involutions: $i_m\colon P(x)\mapsto (-1)^dP(-x)$ and $i_r\colon P(x)\mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex conjugate pairs of roots ($2n\leq d$);2) for $1\leq s\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $d\geq 6$, we give an example of a set $\mathcal{P}^d_{\sigma ,(\ell^+,\ell^-)}$ having $\Lambda (d)$ connected compo\-nents, where $\Lambda (d)\rightarrow \infty$ as $d\rightarrow \infty$.

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On discrete orthogonal U-Bernoulli Korobov-type polynomials

Urieles,

Ramirez,

Cesarano

2024

Constructive Mathematical Analysis

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The primary objective of this paper is to introduce and examine the new class of discrete orthogonal polynomials called $U$-Bernoulli Korobov-type polynomials. Furthermore, we derive essential recurrence relations and explicit representations for this polynomial class. Most of the results are proven through the utilization of generating function methods. Lastly, we place particular emphasis on investigating the orthogonality relation associated with these polynomials.

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The disconnectedness of certain sets defined after uni-variate polynomials

Cited by 3 publications

References 20 publications

Beyond Descartes’ rule of signs

Beyond Descartes’ rule of signs

Real univariate polynomials with given signs of coefficients and simple real roots

On discrete orthogonal U-Bernoulli Korobov-type polynomials

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