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Vladimir Petrov Kostov

Publications and source records attributed to Vladimir Petrov Kostov.

At least 55 records · Page 3Linked to original sources
arXiv2017

On higher-order discriminants

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$, $n\geq 4$, we consider its higher-order discriminant sets $\{ \tilde{D}_m=0\}$, where $\tilde{D}_m:=$Res$(P,P^{(m)})$, $m=2$, $\ldots$, $n-2$, and their projections in the spaces of the variables $a^k:=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots ,a_n)$. Set $P^{(m)}:=\sum _{j=0}^{n-m}c_ja_jx^{n-m-j}$, $P_{m,k}:=c_kP-x^mP^{(m)}$. We show that Res$(\tilde{D}_m,\partial \tilde{D}_m/\partial a_k,a_k)= A_{m,k}B_{m,k}C_{m,k}^2$, where $A_{m,k}=a_n^{n-m-k}$, $B_{m,k}=$Res$(P_{m,k},P_{m,k}')$ if $1\leq k\leq n-m$ and $A_{m,k}=a_{n-m}^{n-k}$, $B_{m,k}=$Res$(P^{(m)},P^{(m+1)})$ if $n-m+1\leq k\leq n$. The equation $C_{m,k}=0$ defines the projection in the space of the variables $a^k$ of the closure of the set of values of $(a_1,\ldots ,a_n)$ for which $P$ and $P^{(m)}$ have two distinct roots in common. The polynomials $B_{m,k},C_{m,k}\in \mathbb{C}[a^k]$ are irreducible. The result is generalized to the case when $P^{(m)}$ is replaced by a polynomial $P_*:=\sum _{j=0}^{n-m}b_ja_jx^{n-m-j}$, $0\neq b_i\neq b_j\neq 0$ for $i\neq j$.

math.CA↗
arXiv2017

A property of discriminants

For the family $P:=x^n+a_1x^{n-1}+\cdots +a_n$ of complex polynomials in the variable $x$ we study its {\em discriminant} $R:=$Res$(P,P',x)$, $R\in \mathbb{C}[a]$, $a=(a_1,\ldots ,a_n)$. When $R$ is regarded as a polynomial in $a_k$, one can consider its discriminant $\tilde{D}_k:=$Res$(R,\partial R/\partial a_k,a_k)$. We show that $\tilde{D}_k=c_k(a_n)^{d(n,k)}M_k^2T_k^3$, where $c_k\in \mathbb{Q}^*$, $d(n,k):=\min (1,n-k)+\max (0,n-k-2)$, the polynomials $M_k,T_k\in \mathbb{C}[a^k]$ have integer coefficients, $a^k=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots ,a_n)$, the sets $\{ M_k=0\}$ and $\{ T_k=0\}$ are the projections in the space of the variables $a^k$ of the closures of the strata of the variety $\{ R=0\}$ on which $P$ has respectively two double roots or a triple root. Set $P_k:=P-xP'/(n-k)$ for $1\leq k\leq n-1$ and $P_n:=P'$. One has $T_k=${\rm Res}$(P_k,P_k',x)$ for $k\neq n-1$ and $T_{n-1}=${\rm Res}$(P_{n-1},P_{n-1}',x)/a_n$.

math.CA↗
arXiv2016

Some facts about discriminants

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$ we ask the questions at which points its discriminant set can be considered as the graph of a function of all coefficients $a_j$ but one and how its subset of points, where the discriminant set is not smooth, projects on the different coordinate hyperplanes in the space of the coefficients~$a$.

math.CA↗
arXiv2016

Uniform bounds on locations of zeros of partial theta function

We consider the partial theta function $θ(q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<δ_0<δ<1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with $δ_0\leq |q|\leq δ$ and for any $n\geq n_0$ the function $θ$ has exactly $n$ zeros with modulus $<|q|^{-n-1/2}$ counted with multiplicity.

math.CV↗
arXiv2016

On the multiple zeros of a partial theta function

We consider the partial theta function $θ(q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. We show that, for any fixed $q$, if $ζ$ is a multiple zero of the function $θ(q,.)$, then $|ζ|\leq 8^{11}$.

math.CV↗
arXiv2015

On the double zeros of a partial theta function

The series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $q\in [0,1)$, $x\in \mathbb{R}$, and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. For $q$ taking one of the {\em spectral} values $\tilde{q}_1$, $\tilde{q}_2$, $\ldots$ (where $0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1$, $\lim _{j\rightarrow \infty}\tilde{q}_j=1$) the function $θ(q,.)$ has a double zero $y_j$ which is the rightmost of its real zeros (the rest of them being simple). For $q\neq \tilde{q}_j$ the partial theta function has no multiple real zeros. We prove that $\tilde{q}_j=1-π/2j+(\log j)/8j^2+O(1/j^2)$ and $y_j=-e^πe^{-(\log j)/4j+O(1/j)}$.

math.CA↗
arXiv2015

On a partial theta function and its spectrum

The bivariate series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ %(where $(q,x)\in {\bf C}^2$, $|q|<1$) defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $θ(q,.)$ is an entire function. For $q\in (-1,0)$ the function $θ(q,.)$ has infinitely many negative and infinitely many positive real zeros. There exists a sequence $\{ \bar{q}_j\}$ of values of $q$ tending to $-1^+$ such that $θ(\bar{q}_k,.)$ has a double real zero $\bar{y}_k$ (the rest of its real zeros being simple). For $k$ odd (resp. for $k$ even) $θ(\bar{q}_k,.)$ has a local minimum at $\bar{y}_k$ and $\bar{y}_k$ is the rightmost of the real negative zeros of $θ(\bar{q}_k,.)$ (resp. $θ(\bar{q}_k,.)$ has a local maximum at $\bar{y}_k$ and for $k$ sufficiently large $\bar{y}_k$ is the second from the left of the real negative zeros of $θ(\bar{q}_k,.)$). For $k$ sufficiently large one has $-1<\bar{q}_{k+1}<\bar{q}_k<0$. One has $\bar{q}_k=1-(π/8k)+o(1/k)$ and $|\bar{y}_k|\rightarrow e^{π/2}=4.810477382\ldots$.

math.CA↗
arXiv2015

Interlacing properties and the Schur-Szegő composition

Each degree $n$ polynomial in one variable of the form $(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable in a unique way as a Schur-Szegő composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, see \cite{Ko1}, \cite{AlKo} and \cite{Ko2}. Set $σ_j:=\sum _{1\leq i_1<\cdots <i_j\leq n-1}a_{i_1}\cdots a_{i_j}$. The eigenvalues of the affine mapping $(c_1,\ldots ,c_{n-1})\mapsto (σ_1,\ldots ,σ_{n-1})$ are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.

math.CA↗
arXiv2015

A mapping defined by the Schur-Szegő composition

Each degree $n+k$ polynomial of the form $(x+1)^k(x^n+c_1x^{n-1}+\cdots +c_n)$, $k\in \mathbb{N}$, is representable as Schur-Szegő composition of $n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the affine mapping $Φ_{n,k}$~:~$(c_1,\ldots ,c_n)$ $\mapsto$ $(σ_1, \ldots ,σ_n)$, where $σ_i$ are the elementary symmetric polynomials of the numbers $a_j$. We study also properties of a similar mapping for functions of the form $e^xP$, where $P$ is a polynomial, $P(0)=1$, and we extend the Descartes rule to them.

math.CA↗
arXiv2015

A property of a partial theta function

The series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $|q|<1$ and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. It is known that for $q$ taking one of the {\em spectral} values $\tilde{q}_1$, $\tilde{q}_2$, $\ldots$ (where $0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1$, $\lim _{j\rightarrow \infty}\tilde{q}_j=1$) the function $θ(q,.)$ has a double zero which is the rightmost of its real zeros (the rest of them being simple). For $q\neq \tilde{q}_j$ the partial theta function has no multiple real zeros. We prove that: 1) for $q\in (\tilde{q}_{j},\tilde{q}_{j+1}]$ the function $θ$ is a product of a degree $2j$ real polynomial without real roots and a function of the Laguerre-Pólya class $\cal{LP-I}$; 2) for $q\in \mathbb{C}\backslash 0$, $|q|<1$, $θ(q,x)=\prod _i(1+x/x_i)$, where $-x_i$ are the zeros of $θ$; 3) for any fixed $q\in \mathbb{C}\backslash 0$, $|q|<1$, the function $θ$ has at most finitely-many multiple zeros; 4) for any $q\in (-1,0)$ the function $θ$ is a product of a real polynomial without real zeros and a function of the Laguerre-Pólya class $\cal{LP}$. 5) for any fixed $q\in \mathbb{C}\backslash 0$, $|q|<1$, and for $k$ sufficiently large, the function $θ$ has a zero $ζ_k$ close to $-q^{-k}$. These are all but finitely-many of the zeros of $θ$.

math.CA↗
arXiv2015

A refined realization theorem in the context of the Schur-Szegő composition

Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szegő composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the $8$-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, where $R$ is a polynomial.

math.CA↗
arXiv2015

Asymptotic expansions of zeros of a partial theta function

The bivariate series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $θ(q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in $q$ of the zeros of $θ$. The coefficients $r_k$ of the stabilized series are positive integers. They are the elements of a known increasing sequence satisfying the recurrence relation $r_k=\sum _{ν=1}^{\infty}(-1)^{ν-1}(2ν+1)r_{k-ν(ν+1)/2}$.

math.CA↗
arXiv2015

On the spectrum of a partial theta function

The bivariate series $θ(q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$, $θ(q,.)$ is an entire function. We show that for $|q|\leq 0.108$ the function $θ(q,.)$ has no multiple zeros.

math.CA↗
arXiv2011

Hardy-Petrovitch-Hutchinson's problem and partial theta function

In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.

math.CA↗
arXiv2003

On the Deligne-Simpson problem and its weak version

We consider the {\em Deligne-Simpson problem (DSP) (resp. the weak DSP): Give necessary and sufficient conditions upon the choice of the $p+1$ conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples (resp. $(p+1)$-tuples with trivial centralizers) of matrices $A_j\in c_j$ with zero sum or of matrices $M_j\in C_j$ whose product is $I$.} The matrices $A_j$ (resp. $M_j$) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.

math.AG↗
arXiv2003

The connectedness of some varieties and the Deligne-Simpson problem

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ or $c_j\subset gl(n,{\bf C})$ so that there exist irreducible (resp. with trivial centralizer) $(p+1)$-tuples of matrices $M_j\in C_j$ or $A_j\in c_j$ satisfying the equality $M_1... M_{p+1}=I$ or $A_1+... +A_{p+1}=0$}. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. For $(p+1)$-tuples of conjugacy classes one of which is with distinct eigenvalues 1) we prove that the variety $\{(M_1,..., M_{p+1})|M_j\in C_j,M_1... M_{p+1}=I\}$ or $\{(A_1,..., A_{p+1})|A_j\in c_j,A_1+... +A_{p+1}=0\}$ is connected if the DSP is positively solved for the given conjugacy classes and 2) we give necessary and sufficient conditions for the positive solvability of the weak DSP.

math.AG↗
arXiv2002

On arrangements of the roots of a hyperbolic polynomial and of one of its derivatives

We consider real monic {\em hyperbolic} polynomials in one real variable, i.e. polynomials having only real roots. Call {\em hyperbolicity domain} $Π$ of the family of polynomials $P(x,a)=x^n+a_1x^{n-1}+... +a_n$, $a_i,x\in {\bf R}$, the set $\{a\in {\bf R}^n| P $is hyperbolic $\}$. The paper studies a stratification of $Π$ defined by the arrangement of the roots of $P$ and $P^{(k)}$, where $2\leq k\leq n-1$. We prove that the strata are smooth contractible real algebraic varieties.

math.AG↗