For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q\left(m\right)$, where $Q$ is a polynomial with integral coefficients, then $P\left(x\right)=Q\left(R\right(x\left)\right)$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P\left(x\right)={\left(R\left(x\right)\right)}^q$ for some polynomial $R\left(x\right)$.

A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_1$ and $D_2$ such that $A^{-\mathrm{T}}=D_{1}A{D}_2$, where $A^{-\mathrm{T}}$ denotes the transpose of the inverse of $A$. Denote by $J=\mathrm{diag}(\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^{\mathrm{T}}JQ=J$. Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

Let $A$, $B$ be $n\times m$ matrices. The concept of matrix majorization means the $j$th column of $A$ is majorized by the $j$th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$. We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

Let ${\mathbb{Z}}_{+}$ be the semiring of all nonnegative integers and $A$ an $m\times n$ matrix over ${\mathbb{Z}}_{+}$. The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m\times k$ matrix times a $k\times n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2\times 2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$. For $A$ with isolation number $k$, we investigate the possible values of the...

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${\left\{P_i\right\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i\left(x\right)+Q_1\left(x\right)P_{i-1}\left(x\right)+Q_2\left(x\right)P_{i-2}\left(x\right)=0$ with the standard initial conditions $P_0\left(x\right)=1,P_{-1}\left(x\right)=0,$ where $Q_1\left(x\right)$ and $Q_2\left(x\right)$ are arbitrary real polynomials.

A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A\otimes x=x$. An eigenvector $x$ of $A$ is called the greatest $𝐗$-eigenvector of $A$ if $x\in 𝐗=\{x;\underline{x}\le x\le \overline{x}\}$ and $y\le x$ for each eigenvector $y\in 𝐗$. A max-min matrix $A$ is called strongly $𝐗$-robust if the orbit $x,A\otimes x,A^2\otimes x,\cdots$ reaches the greatest $𝐗$-eigenvector with any starting vector of $𝐗$. We suggest an $O\left(n^3\right)$ algorithm for computing the greatest $𝐗$-eigenvector of $A$ and study the strong $𝐗$-robustness. The necessary and sufficient conditions for strong $𝐗$-robustness are introduced...

A $0/1$-simplex is the convex hull of $n+1$ affinely independent vertices of the unit $n$-cube $I^n$. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0/1$-simplices in $I^n$ can be represented by $0/1$-matrices $P$ of size $n\times n$ whose Gramians $G=P^{\top }P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{-\top }$ of $P$ is doubly stochastic and has the...

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $\left(J_{\infty }\right)$: There does not exist a sequence $(p_k,q_k)\subset ℂ²$ of complex singular points of F such that the imaginary parts $(\Im \left(p_k\right),\Im \left(q_k\right))$ tend to (0,0), the real parts $(\Re \left(p_k\right),\Re \left(q_k\right))$ tend to ∞ and $F(\Re \left(p_k\right),\Re \left(q_k\right)))\to a\in ℝ²$. It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $\left(J_{\infty }\right)$ and if, in addition, the restriction of F to every real level set $P^{-1}\left(c\right)$ is proper for values of |c| large enough.

An $m\times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let ${𝐌}_{m,n}$ be the set of all $m\times n$ real matrices. For $A,B\in {𝐌}_{m,n}$, we say that $A$ is row Hadamard majorized by $B$ (denoted by $A{\prec }_{RH}B)$ if there exists an $m\times n$ row stochastic matrix $R$ such that $A=R\circ B$, where $X\circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in {𝐌}_{m,n}$. In this paper, we consider the concept of row Hadamard majorization as a relation on ${𝐌}_{m,n}$ and characterize the structure of all linear operators $T:{𝐌}_{m,n}\to {𝐌}_{m,n}$ preserving...

A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A=B{B}^{\top }$. If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$. In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$. Failure of this algorithm implies that $A$ does not have cp-rank $3$. Our motivation stems from the question if there exist three nonnegative polynomials of degree at...

Let $m$ and $n$ be positive integers, and let $R=(r_1,...,r_m)$ and $S=(s_1,...,s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m\times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F\left(R\right)$ be the $m\times n$ $(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\mathrm{disc}\left(A\right)$, is the number of positions in which $F\left(R\right)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over...

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 $\mathrm{pos}\le c$ positive and $\neg \le p$ negative roots, where $\mathrm{pos}\equiv c(mod2)$ and $\neg \equiv p(mod2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\mathrm{pos},\mathrm{neg})$ satisfying these conditions there exists a polynomial $P$ with exactly $\mathrm{pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\ge 4$...

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ${ₙ}^c$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ\in {ₙ}^c$ of the form $Pₙ\left(z\right):={\sum }_{j=0}^n{a}_j{z}_j$, $a_j\in C$, by $Sₙ\left(Pₙ\right)\left(z\right):={\sum }_{j=0}^{n}a{̃}_j{z}_j$, $a{̃}_j:=a_j\left|a_j\right|min|a_j|,1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{real}:=su{p}_{Pₙ\in ₙ}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|)$, $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{comp}:=su{p}_{Pₙ\in {ₙ}^c}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

Let ${𝕄}_{n,m}$ be the set of all $n\times m$ real or complex matrices. For $A,B\in {𝕄}_{n,m}$, we say that $A$ is row-sum majorized by $B$ (written as $A{\prec }^{\mathrm{rs}}B$) if $R\left(A\right)\prec R\left(B\right)$, where $R\left(A\right)$ is the row sum vector of $A$ and $\prec$ is the classical majorization on ${ℝ}^n$. In the present paper, the structure of all linear operators $T:{𝕄}_{n,m}\to {𝕄}_{n,m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on ${ℝ}^n$ and then find the linear preservers of row-sum majorization of these relations on ${𝕄}_{n,m}$. ...