arXiv subjects

Petar P. Petrov

Publications and source records attributed to Petar P. Petrov.

3 recordsLinked to original sources

arXiv2024

A polynomial-time algorithm for deciding the Hilbert Nullstellensatz over $\mathbb{Z}_2$. A proof of $\mathbf{P}=\mathbf{NP}$ hypothesis

Let ${\mathbf P}$ be the class of polynomial-time decision problems and $\mathbf{NP}$ be the class of nondeterministic polynomial time decision problems. We prove the following: Theorem 3. The classes ${\mathbf P}$ and $\mathbf{NP}$ are equivalent. That is, ${\mathbf P}=\mathbf{NP}$. Theorem 3 gives a positive answer to the question $$\hbox{Does }{\mathbf P}=\mathbf{NP}?,$$ see S. Cook, The $\mathbf{P}$ versus $\mathbf{NP}$ problem, Official problem description, www.claymath.org/millennium-problems. Crucial for its proof is Theorem 2, from which it follows that the $\mathbf{NP}$-complete problem of deciding the Hilbert Nullstellensatz over $\mathbb{Z}_2$ belongs to the class ${\mathbf P}$. Theorem 2. There is a constructive algorithm for deciding the Hilbert Nullstellensatz over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ is the space of all complex numbers with integer real and imaginary parts. The number $s(n,m_σ)$ of basic steps of the algorithm, where $n$ is the number of variables and $m_σ$ is the total length of input polynomials, satisfies the inequality \begin{eqnarray*} & & s(n,m_σ) \\ & \le & c_2\{m_σ^2\log m_σ+\min\{[m_σ^{(1)}]^3,(d_1)^3\}+\sum_{\ell =1}^{n-2}N^{(l)}\min\{[m_σ^{(\ell +1)}]^2,(d_{\ell +1})^2)\}\\ && +N^{(n-1)}\min \{m_σ,d_n\} \} \end{eqnarray*} where $c_2$ is an absolute constant, $\{d_{\ell}\}_{\ell=1}^n$ are the maximal partial degrees in $\{z_{\ell}\}_{\ell=1}^n$, respectively, and the numbers $m_σ^{(\ell)}$ and $N^{(\ell)}$ are characteristics of the input polynomials, concerning partial lengths and numbers of major sub-monomials it the natural order of monomials, defined in the body of the paper.

math.GM↗

arXiv2021

Analyticity domains of critical points of polynomials. A proof of Sendov's conjecture

Let $¶_{n}^c(\barμ,\barν)$ be the set of all complex polynomials $p(z)=\prod_{i=1}^{m}(z-z_i)^{μ_i}$, $\sum_{i=1}^mμ_i=n$, with derivatives of the form $$ p'(z)=n\prod_{i=1}^{m}(z-z_i)^{μ_i-1}\prod_{j=1}^{k}(z-ξ_j)^{ν_j}, ~\sum_{j=1}^kν_j=m-1. $$ In this note we prove the following:\par\medskip {\it \noindent For a fixed ordering $α=(1,2,\ldots,m)$, the distinct zeros $\{z_i\}_{i=1}^m$ and the distinct critical points of the second kind $\{ξ_j\}_{j=1}^k$ of polynomials from $¶_{n}^c(\barμ,\barν)$ are analytic functions $\{z_i^{αβ}\}_{i=1}^m$ and $\{ξ_j^{αβ}\}_{j=1}^k$, resp., $β=(i_1,i_2,\ldots,i_{k+1})$, of any of the variables $(z_{i_1},z_{i_2},\ldots,z_{i_{k+1}})$ in the domain $$ \{(z_{i_1},z_{i_2},\ldots,z_{i_{k+1}})\in \C^{k+1}~\vert~p\in ¶_{n}^c(\barμ,\barν) \}, $$ being also continuous on its boundary.}\par\medskip\noindent This statement gives an immediate proof to the well-known conjecture of Bl. Sendov \cite{sen}: \par\medskip {\it \noindent If $n\ge 2$ and $p(z)=\prod_{i=1}^n (z-z_i)$ is a polynomial of degree $n$ such that $z_i\in \C$, $\vert z_i\vert\le 1$, $i=1,2,\ldots,n$, then for every $i=1,2,\ldots,n$, the disk $\{z\in \C\,|\,\vert z_i-z\vert \le 1\}$ contains at least one zero of $p'(z)$.}

math.CV↗

arXiv2014

Surface roughness evolution in a solid-on-solid model of epitaxial growth

The paper presents results from kinetic Monte Carlo simulations of kinetic surface roughening using an important and experimentally relevant model of epitaxial growth -- the solid-on-solid model with Arrhenius dynamics. A restriction on diffusing adatoms is included allowing hopping down only at steps of height one monolayer in order to avoid possibly unrealistic events of jumping from arbitrarily high steps. Simulation results and precise analytic expressions representing the time evolution of surface roughness do not depend on the substrate size and clearly put forward the conclusion that for any basic set of parameters the model approaches in asymptotic limit the usual random deposition process with growth exponent $β=1/2$. At high temperatures, it is preceded by a long transient regime characterized by a smooth surface covered with porous pillars and described by a power law with $β=3/4$.

cond-mat.mtrl-sci↗