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Vladimir Petrov Kostov

Publications and source records attributed to Vladimir Petrov Kostov.

At least 37 records · Page 2Linked to original sources
arXiv2020

Hyperbolic polynomials and rigid moduli orders

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. We consider HPs with distinct moduli of the roots. We ask the question when the order of the moduli of the negative roots w.r.t. the positive roots on the real positive half-line completely determines the signs of the coefficients of the polynomial. When there is at least one positive and at least one negative root this is possible exactly when the moduli of the negative roots interlace with the positive roots (hence half or about half of the roots are positive). In this case the signs of the coefficients of the HP are either $(+,+,-,-,+,+,-,-,\ldots )$ or $(+,-,-,+,+,-,-,+,\ldots )$.

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arXiv2020

Hyperbolic polynomials and canonical sign patterns

A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. We discuss the question: If the moduli of all $c+p$ roots are distinct and ordered on the positive half-axis, then at which positions can the $p$ moduli of negative roots be depending on the positions of the positive and negative signs of the coefficients of the polynomial? We are especially interested in the choices of these signs for which exactly one order of the moduli of the roots is possible.

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arXiv2020

Degree 5 polynomials and Descartes' rule of signs

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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arXiv2019

A non-realization theorem in the context of Descartes' rule of signs

For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Descartes' rule of signs says that $P$ has $pos\leq c$ positive and $neg\leq p$ negative roots, where $pos\equiv c($\, mod $2)$ and $neg\equiv p($\, mod $2)$. For $1\leq d\leq 3$, 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 $neg$ negative roots (all of them simple); that is, all these cases are realizable. This is not true for $d\geq 4$, yet for $4\leq d\leq 8$ (for these degrees the exhaustive answer to the question of realizability is known) in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\geq 4$. For $d=9$, we show a counterexample to this conjecture: for the sign pattern $(+,-,-,-,-,+,+,+,+,-)$ and the pair $(1,6)$ there exists no polynomial with $1$ positive, $6$ negative simple roots and a complex conjugate pairs and, up to equivalence, this is the only case for $d=9$.

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arXiv2019

Two properties of the partial theta function

For the partial theta function $θ(q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is a simple or double zero of $θ(q^{\flat},.)$. For $q\in (0,1)$, $q\rightarrow 1^-$ and $a\geq e^π$, there are $o(1/(1-q))$ and $(\ln (a/e^π))/(1-q)+o(1/(1-q))$ real zeros of $θ(q,.)$ in the intervals $[-e^π,0)$ and $[-a,-e^{-π}]$ respectively (and none in $[0,\infty)$). For $q\in (-1,0)$, $q\rightarrow -1^+$ and $a\geq e^{π/2}$, there are $o(1/(1+q))$ real zeros of $θ(q,.)$ in the interval $[-e^{π/2},e^{π/2}]$ and $(\ln (a/e^{π/2})/2)/(1+q)+o(1/(1+q))$ in each of the intervals $[-a,-e^{π/2}]$ and $[e^{π/2},a]$.

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arXiv2019

Moduli of roots of hyperbolic polynomials and Descartes' rule of signs

A real univariate polynomial with all roots real is called hyperbolic. By Descartes' rule of signs for hyperbolic polynomials (HPs) with all coefficients nonvanishing, a HP with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients has exactly $c$ positive and $p$ negative roots. For $c=2$ and for degree $6$ HPs, we discuss the question: When the moduli of the $6$ roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its two positive roots depending on the positions of the two sign changes in the sequence of coefficients?

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arXiv2019

On Descartes' rule of signs

A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum _{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $σ:=(+$,sgn$(a_{d-1})$, $\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of positive (resp. $neg$ of negative) roots of $P$, one has $pos\leq c$ (resp. $neg\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign preservations in $σ$; the numbers $c-pos$ and $p-neg$ are even. We say that $P$ realizes the SP $σ$ with the pair $(pos, neg)$. For SPs with $c=2$, we give some sufficient conditions for the (non)realizability of pairs $(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\ldots$, $[(d-2)/2]$.

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arXiv2019

Descartes' rule of signs and moduli of roots

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. For $c=1$ and $2$, we discuss the question: When the moduli of all the roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its positive roots depending on the positions of the sign changes in the sequence of coefficients?

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arXiv2019

On the complex conjugate zeros of the partial theta function

We prove that 1) for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ belong to the set $\{$~Re\,$x\in (-5792.7,0),$~$|$Im\,$x|<132~\}$ $\cup$ $\{ ~|x|<18~\}$ and 2) for any $q\in (-1,0)$, they belong to the rectangle $\{$~$|$Re\,$x|< 364.2,$~$|$Im\,$x|<132~\}$.

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arXiv2019

New aspects of Descartes' rule of signs

Below we summarize some new developments in the area of distribution of roots and signs of real univariate polynomials pioneered by R.~Descartes in the middle of the 17-th century.

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arXiv2018

Addendum: A separation in modulus property of the zeros of a partial theta function

We consider the partial theta function $θ(q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $D(a):=\{ q\in \mathbb{C}$, $0<|q|\leq a$, $\arg (q)\in [π/2,3π/2]\}$. We show that for $k\in \mathbb{N}$ and $q\in D(0.55)$, there exists exactly one zero of $θ(q,.)$ (which is a simple one) in the open annulus $|q|^{-k+1/2}<z<|q|^{-k-1/2}$ (if $k\geq 2$) or in the punctured disk $0<z<|q|^{-3/2}$ (if $k=1$). For $k=1$, $4$, $5$, $6$, $\ldots$, this holds true for $q\in D(0.6)$ as well.

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arXiv2018

On the zero set of the partial theta function

We consider the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $q\in (-1,0)\cup (0,1)$ and either $x\in \mathbb{R}$ or $x\in \mathbb{C}$. We prove that for $x\in \mathbb{R}$, in each of the two cases $q\in (-1,0)$ and $q\in (0,1)$, its zero set consists of countably-many smooth curves in the $(q,x)$-plane each of which (with the exception of one curve for $q\in (-1,0)$) has a single point with a tangent line parallel to the $x$-axis. These points define double zeros of the function $θ(q,.)$; their $x$-coordinates belong to the interval $[-38.83\ldots ,-e^{1.4}=4.05\ldots )$ for $q\in (0,1)$ and to the interval $(-13.29,23.65)$ for $q\in (-1,0)$. For $q\in (0,1)$, infinitely-many of the complex conjugate pairs of zeros to which the double zeros give rise cross the imaginary axis and then remain in the half-disk $\{ |x|<18$, Re\,$x>0\}$. For $q\in (-1,0)$, complex conjugate pairs do not cross the imaginary axis.

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arXiv2018

Descartes' rule of signs, Rolle's theorem and sequences of admissible pairs

Given a real univariate degree $d$ polynomial $P$, the numbers $pos_k$ and $neg_k$ of positive and negative roots of $P^{(k)}$, $k=0$, $\ldots$, $d-1$, must be admissible, i.e. they must satisfy certain inequalities resulting from Rolle's theorem and from Descartes' rule of signs. For $1\leq d\leq 5$, we give the answer to the question for which admissible $d$-tuples of pairs $(pos_k$, $neg_k)$ there exist polynomials $P$ with all nonvanishing coefficients such that for $k=0$, $\ldots$, $d-1$, $P^{(k)}$ has exactly $pos_k$ positive and $neg_k$ negative roots all of which are simple.

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arXiv2017

A domain containing all zeros of the partial theta function

We consider the partial theta function, i.e. the sum of the bivariate series $θ(q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$ for $q\in (0,1)$, $z\in \mathbb{C}$. We show that for any value of the parameter $q\in (0,1)$ all zeros of the function $θ(q,.)$ belong to the domain $\{ {\rm Re}~z<0, |{\rm Im}~z|\leq 132\}$$\cup$$\{ {\rm Re}~z\geq 0, |z|\leq 18\}$.

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arXiv2017

Polynomials, sign patterns and Descartes' rule of signs

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 $pos\leq c$ positive and $neg\leq p$ negative roots, where $pos\equiv c($\, mod $2)$ and $neg\equiv p($\, mod $2)$. For $1\leq d\leq 3$, 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 $d\geq 4$ this is not so. It was observed that for $4\leq d\leq 10$, in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\geq 4$. 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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arXiv2017

A separation in modulus property of the zeros of a partial theta function

We consider the partial theta function $θ(q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $α_0~:=~\sqrt{3}/2π~=~0.2756644477\ldots$. We show that, for $n\geq 5$, for $|q|\leq 1-1/(α_0n)$ and for $k\geq n$ there exists a unique zero $ξ_k$ of $θ(q,.)$ satisfying the inequalities $|q|^{-k+1/2}<|ξ_k|<|q|^{-k-1/2}$; all these zeros are simple ones. The moduli of the remaining $n-1$ zeros are $\leq |q|^{-n+1/2}$. A {\em spectral value} of $q$ is a value for which $θ(q,.)$ has a multiple zero. We prove the existence of the spectral values $0.4353184958\ldots \pm i\, 0.1230440086\ldots$ for which $θ$ has double zeros $-5.963\ldots \pm i\, 6.104\ldots$.

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arXiv2017

Something You Always Wanted to Know About Real Polynomials (But Were Afraid to Ask)

The famous Descartes' rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomials in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration for generations of mathematicians. Trying to extend and sharpen this rule, we consider below the set of all real univariate polynomials of a given degree, a given collection of signs of their coefficients, and a given number of positive and negative roots. In spite of the elementary definition of the main object of our study, it is a non-trivial question for which sign patterns and numbers of positive and negative roots the corresponding set is non-empty. The main result of the present paper is a discovery of a new infinite family of non-realizable combinations of sign patterns and the numbers of positive and negative roots.

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arXiv2017

On realizability of sign patterns by real polynomials

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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