Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of...

We show that for a polynomial mapping $F=(f₁,...,fₘ):{ℂ}^n\to {ℂ}^m$ the Łojasiewicz exponent ${}_{\infty }\left(F\right)$ of F is attained on the set $z\in {ℂ}^n:f₁\left(z\right)·...·fₘ\left(z\right)=0$.

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

We study the extensibility of piecewise polynomial functions defined on closed subsets of ${ℝ}^2$ to all of ${ℝ}^2$. The compact subsets of ${ℝ}^2$ on which every piecewise polynomial function is extensible to ${ℝ}^2$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁\left(x\right)=\cdots =h_r\left(x\right)=0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f|}_V$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h\left(x\right)={\sum }_{i=1}^{r}h{²}_i\left(x\right){\sigma }_i\left(x\right)$, where ${\sigma }_i$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p,w}\left(u\right)$ are defined by $q_{p,w}\left(u\right)\equiv (u^w-u^{wp})/pmodp$ with $0\le q_{p,w}\left(u\right)\le p-1$, u ≥ 0, which are generalizations of Fermat quotients $q_{p,p-1}\left(u\right)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f\left(u\right)\equiv q_{p,w}\left(u\right)modp$ for a given polynomial f(x) over the finite field ${}_p$. In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...

In the space ${\Lambda }^p$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^p$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that ${\Lambda }^p$ splits as the direct sum $d*\left({\Lambda }^{p+1}\right)\oplus \delta *\left({\Lambda }^{p-1}\right)\oplus H^p$, where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

We consider the space ${}^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on ${}^{ℳ}$. A description of the space ${(}^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

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.

Let $f=ax+b{x}^q+x^{2q-1}{\in }_q\left[x\right]$. We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of ${}_{q²}$. This result allows us to solve a related problem: Let $g_{n,q}{\in }_p\left[x\right]$ (n ≥ 0, $p=cha{r}_q$) be the polynomial defined by the functional equation ${\sum }_{c{\in }_q}{(x+c)}^n=g_{n,q}(x^q-x)$. We determine all n of the form $n=q^{\alpha }-q^{\beta }-1$, α > β ≥ 0, for which $g_{n,q}$ is a permutation polynomial of ${}_{q²}$.

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 K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials $u₀,...,u_{m-1}\in K[X₀,...,X_{m-1}]$ such that $f\left({\sum }_{j=0}^{m-1}{\xi }^j{X}_j\right)={\sum }_{j=0}^{m-1}{\xi }^j{u}_j$. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u₀,...,u_{m-1}$ have no common divisor in $K[X₀,...,X_{m-1}]$ of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several...

We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f\left(x\right),g\left(x\right){\in }_p\left[x\right]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^{1/E(1+\kappa /(1-\kappa )}>M>p^{\epsilon }$, where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $\left|f\left([0,M]\right)\cap g\left([0,M]\right)\right|\lesssim M^{1-\epsilon }$.

Let $P\left(z\right)$ be a polynomial of degree $n$ having no zeros in $\left|z\right|<k$, $k\le 1$, and let $Q\left(z\right):=z^n\overline{P(1/\overline{z})}$. It was shown by Govil that if ${max}_{\left|z\right|=1}\left|P^{\text{'}}\left(z\right)\right|$ and ${max}_{\left|z\right|=1}\left|Q^{\text{'}}\left(z\right)\right|$ are attained at the same point of the unit circle $\left|z\right|=1$, then $$\underset{\left|z\right|=1}{max}|P^{\text{'}}\left(z\right)|\le \frac{n}{1+k^n}\underset{\left|z\right|=1}{max}\left|P\left(z\right)\right|.$$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

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...

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $𝔽S_n$ of the symmetric group $S_n$ over a field $𝔽$ of characteristic 0 (or $p>n$). The goal is to obtain a constructive version of the isomorphism $\psi :{⨁}_{\lambda }{M}_{d_{\lambda }}\left(𝔽\right)⟶𝔽S_n$ where $\lambda$ is a partition of $n$ and $d_{\lambda }$ counts the standard tableaux...