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Vladimir Petrov Kostov

Publications and source records attributed to Vladimir Petrov Kostov.

At least 19 recordsLinked to original sources
arXiv2026

Where not to find the spectrum of the partial theta function

The spectrum of Ramanujan's partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $θ(q,.)$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup \mathbb{D}_{c_0}$, $c_0=0.2078750206\ldots$, where $\mathbb{S}$ is the sector $\{ 0<|z|<0.6,{\rm arg}(z)\in [π/4 ,7π/4 ]\}$. There is a single spectral value in the set $\mathbb{S}\cup \mathbb{D}_{0.31}$ which equals $0.309249\ldots$. For $q\in \mathbb{S}\cup \mathbb{D}_{c_0}$, the moduli of the zeros of $θ$ are separated by the negative half-integer powers of $|q|$.

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arXiv2026

Some analytic properties of the partial theta function

We prove new properties of the zero set of Ramanujan's partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\geq 5$, such that all real zeros of $θ(q,.)$ lie to its left and all complex zeros to its right. A similar property is proved for $q\in (-1,0)$. For $q\in (0,1)$, there are no real zeros $\geq -6$. For $q\in (-1,0)$, there are no negative zeros $\geq -2.4$ and no positive zeros $\leq 2.4$, except the smallest one.

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arXiv2026

Three realization problems about univariate polynomials

We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\sum_{j=0}^db_jx^j$ defines the sign pattern $σ(P):=({\rm sgn}(b_d)$, $\ldots$, ${\rm sgn}(b_0))$. The numbers $p_d$ and $n_d$ of positive and negative roots of $P$ (counted with multiplicity) satisfy the Descartes' rule of signs. Problem~1 asks for which couples $C$ of the form (sign pattern $σ$, pair $(p_d,n_d)$ compatible with $σ$ in the sense of Descartes' rule of signs), there exist polynomials $P$ defining these couples. Problem~2 asks for which $d$-tuples of pairs $T:=((p_d,n_d)$, $\ldots$, $(p_1,n_1))$, there exist polynomials $P$ such that $P^{(d-j)}$ has $p_j$ positive and $n_j$ negative roots. A $d$-tuple $T$ determines the sign pattern $σ(P)$, but the inverse is false. We show by an example that $6$ is the smallest value of $d$ for which there exist non-realizable tuples $T$ for which the corresponding couples $C$ are realizable. The third problem concerns polynomials with all roots real. We give a geometric interpretation of the three problems in the context of degree $4$ polynomials.

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arXiv2025

Midpoints and critical points

For a degree $5$ real polynomial with roots $x_1\leq \cdots \leq x_5$ and roots $ξ_1\leq \cdots \leq ξ_4$ of its derivative, we set $z_j:=(x_j+x_{j+1})/2$, $1\leq j\leq 4$. We prove that one cannot have at the same time $\min_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \min_{1\leq j\leq 3}(ξ_{j+1}-ξ_j)$ and $\max_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \max_{1\leq j\leq 3}(ξ_{j+1}-ξ_j)$. The result settles a general question about midpoints and critical points of hyperbolic polynomials.

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arXiv2025

On the location of the complex conjugate zeros of the partial theta function

We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ with non-negative real part belong to the half-annulus $\{$Re$(x)\geq 0,~1<|x|<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{π/2}=4.810477382\ldots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\ldots ]$, $θ(q,.)$ has no zeros with non-negative real part. The complex conjugate pairs of zeros with negative real part belong to the left open half-disk of radius $49.8$ centered at the origin.

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arXiv2024

On universal sign patterns

We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $σ(Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the polynomial $Q$ has $\tilde{c}$ positive and $d-\tilde{c}$ negative roots. We suppose the moduli of these roots distinct. The {\em order} of these moduli is defined when in their string as points of the positive half-axis one marks the places of the moduli of negative roots. A sign pattern $σ^0$ is {\em universal} when for any possible order of the moduli there exists a polynomial $Q$ with $σ(Q)=σ^0$ and with this order of the moduli of its roots. We show that when the polynomial $P_{m,n}:=(x-1)^m(x+1)^n$ has no vanishing coefficients, the sign pattern $σ(P_{m,n})$ is universal. We also study the question when $P_{m,n}$ can have vanishing coefficients.

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arXiv2024

On a problem inspired by Descartes' rule of signs

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative followed by $q$ positive coefficients. We consider the sequence of moduli of their roots on the positive real half-axis; all moduli are supposed distinct. We mark in this sequence the positions of the moduli of the two positive roots. For $m=n=2$, $n=q=2$ and $m=q=2$, we give the exhaustive answer to the question which the positions of the two moduli of positive roots can be.

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arXiv2023

Degree $6$ hyperbolic polynomials and orders of moduli

We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\tilde{c}$ positive and $\tilde{p}$ negative roots counted with multiplicity, where $\tilde{c}$ and $\tilde{p}$ are the numbers of sign changes and sign preservations in the sequence of its coefficients, $\tilde{c}+\tilde{p}=d$. For $d=6$, we give the exhaustive answer to the question: When the moduli of all $6$ roots are distinct and arranged on the real positive half-axis, in which positions can the moduli of the negative roots be depending on the signs of the coefficients?

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arXiv2023

On Descartes' rule of signs for hyperbolic polynomials

We consider univariate real polynomials with all roots real and with two sign changes in the sequence of their coefficients which are all non-vanishing. One of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree $d$ polynomials have $2$ positive and $d-2$ negative roots. We consider the sequences of the moduli of their roots on the real positive half-axis. When the moduli are distinct, we give the exhaustive answer to the question at which positions can the moduli of the two positive roots be.

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arXiv2023

Beyond Descartes' rule of signs

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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arXiv2023

Further than Descartes' rule of signs

The {\em sign pattern} defined by the real polynomial $Q:=Σ_{j=0}^da_jx^j$, $a_j\neq 0$, is the string $σ(Q):=({\rm sgn(}a_d{\rm )},\ldots ,{\rm sgn(}a_0{\rm )})$. 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 {\em 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 $pos\geq 1$, $neg\geq 1$, 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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arXiv2022

A domain free of the zeros of the partial theta function

We prove that for $q\in (0,1)$, the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed domain $\{ \{ |x|\leq 3\} \cap \{${\rm Re}$x\leq 0\} \cap \{ |${\rm Im}$x|\leq 3/\sqrt{2}\} \} \subset \mathbb{C}$ and no real zeros $\geq -5$.

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arXiv2022

The disconnectedness of certain sets defined after uni-variate polynomials

We consider the set of monic real univariate 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 $d=6$ and for signs of the coefficients $(+,-,+,+,+,-,+)$, we prove that the set of such polynomials having two positive, two 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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arXiv2021

Univariate polynomials and the contractibility of certain sets

We consider the set $Π^*_d$ of monic polynomials $Q_d=x^d+\sum _{j=0}^{d-1}a_jx^j$, $x\in \mathbb{R}$, $a_j\in \mathbb{R}^*$, having $d$ distinct real roots, and its subsets defined by fixing the signs of the coefficients $a_j$. We show that for every choice of these signs, the corresponding subset is non-empty and contractible. A similar result holds true in the cases of polynomials $Q_d$ of even degree $d$ and having no real roots or of odd degree and having exactly one real root. For even $d$ and when $Q_d$ has exactly two real roots which are of opposite signs, the subset is contractible. For even $d$ and when $Q_d$ has two positive (resp. two negative) roots, the subset is contractible or empty. It is empty exactly when the constant term is positive, among the other even coefficients there is at least one which is negative, and all odd coefficients are positive (resp. negative).

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arXiv2021

Descartes' rule of signs, canonical sign patterns and rigid orders of moduli

We consider real polynomials in one variable without vanishing coefficients and with all roots real and of distinct moduli. We show that the signs of the coefficients define the order of the moduli of the roots on the real positive half-line exactly when no four consecutive signs of coefficients equal $(+,+,-,-)$, $(-,-,+,+)$, $(+,-,-,+)$ or $(-,+,+,-)$.

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arXiv2020

Sign patterns and rigid moduli orders

We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\sum_{j=0}^{d-1}a_jx^j$ and its {\em hyperbolicity domain} $Π_d$, i.e. the subset of values of the coefficients $a_j$ for which the polynomial $Q_d$ has all roots real. The subset $E_d\subset Π_d$ is the one on which a modulus of a negative root of $Q_d$ is equal to a positive root of $Q_d$. At a point, where $Q_d$ has $d$ distinct roots with exactly $s$ ($1\leq s\leq [d/2]$) equalities between positive roots and moduli of negative roots, the set $E_d$ is locally the transversal intersection of $s$ smooth hypersurfaces. At a point, where $Q_d$ has two double opposite roots and no other equalities between moduli of roots, the set $E_d$ is locally the direct product of $\mathbb{R}^{d-3}$ and a hypersurface in $\mathbb{R}^3$ having a Whitney umbrella singularity. For $d\leq 4$, we draw pictures of the sets $Π_d$ and~$E_d$.

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