Narrow Results

We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qⁿ over the finite field of size q. When d is constant, the algorithm needs 𝒪(n^d-1) quantum queries, which is optimal. Classically, Ω(n^d) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.]. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.] also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is 0. In [The reconstruction theorem for endomorphisms, Bull. Braz. Math. Soc. (N.S.) 33(2) (2002) 231–262.] Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the (2m + 1)th image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: “Given a generic polynomial$P$ of degree at least$2,$ there exist positive integers$N,M$ such that$\text{Re}(P^k(z))=\text{Re}(P^k(w))$ for all$0\le k\le N$ implies$P^M(z)=P^M(w)$”.

One of the main challenges in molecular science is to model a chemical complex and predict its thermochemical properties. Various researchers have developed a number of hypothetical techniques in this regard, and one of them is focused on the topological indices. A topological index is a number associated with a molecule’s structural graph that is considered to be capable of predicting certain chemical and physical properties of molecules. Degree-based topological indices are one of the groups of topological indices that are crucial in chemical graph theory. Shannon’s entropy is a fundamental factor of a subclass of topological indices that measures the structural information of the graphs. In this study, we examine the entropy based on topological indices such as the first and second Zagreb indices, the general Randić index, the harmonic and sum-connectivity index of Hanoi graph. Moreover, we determine a number of multiplicative invariants, and different polynomials for Hanoi graphs with graphical illustrations.

In this paper we present a precise description of the connectedness locus of the family of polynomials (z² + x)² + y, where x, y are real numbers.

Applying the averaging theory of first, second and third order to one class generalized polynomial Liénard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit.

We analyze the complexity of deciding the convexity of polynomials over simplexes. For polynomials of degree four or higher, the decision problem requires accomplishing an NP-hard task; when the degrees of the polynomials are less than four, the problem can be solved in NC time.

The aim of this paper is the investigation of derivations in semiring of polynomials over idempotent semiring. For semiring $S[x]$, where $S$ is a commutative idempotent semiring we construct derivations corresponding to the polynomials from the principal ideal $(x)$ and prove that the set of these derivations is a non-commutative idempotent semiring closed under the Jordan product of derivations — Theorem 3.3. We introduce generalized inner derivations defined as derivations acting only over the coefficients of the polynomial and consider $S$-derivations in classical sense of Jacobson. In the main result, Theorem 5.3, we show that any derivation in $S[x]$ can be represented as a sum of a generalized inner derivation and an $S$-derivation.

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and be the unique extension of v to a fixed algebraic closure of K with value group . It is known that a complete distinguished chain for an element θ belonging to with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of , a tower of fields, together with a sequence of elements belonging to which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of satisfying certain properties, it is shown that there exists a complete distinguished chain for an element associated to these invariants. We use the notion of lifting of polynomials to construct it.

Let F be a field and A a F-algebra. The F-linear transformation φ : A → A is a local multiplication map if for all a ∈ A there exists some u_a ∈ A such that φ(a) = au_a. Let denote the F-algebra of all local multiplication maps of A. If F is infinite and F[x] is the ring of polynomials over F, then it is known Lemma 1 in [J. Buckner and M. Dugas, Quasi-Localizations of ℤ, Israel J. Math.160 (2007) 349–370] that . The purpose of this paper is to study for finite fields F. It turns out that in this case is a "very" non-commutative ring of cardinality 2^ℵ₀ with many interesting properties.

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial $P\in \mathbb{N}[x]$, and any directed finite bipartite graph can be considered as a polynomial $P\in \mathbb{N}[x,y]$, and vise verse. We also show that the multiplication in the semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$ corresponds to an operation of the corresponding graphs. This operation is exactly the product of Petri nets in the sense of Winskel [G. Winskel and M. Nielsen, Models of concurrency, in Handbook of Logic in Computer Science, Vol. 4, eds. Abamsky, Gabbay and Maibaum (Oxford University Press, 1995), pp. 1–148]. As an application, we give an approach to dividing in the semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$, and a criteria for parallalization of Petri nets. Finally, we endow the set of all bipartite graphs with the Zariski topology.

We study matrices over arbitrary rings that are integral over the base ring. For any ring $R$, let $f,g,h\in R[x_1,x_2,\dots ,x_n]$ with $h=fg$. We prove that for any pairwise commuting elements $a_1,a_2,\dots ,a_n\in R$ if $g(a_1,a_2,\dots ,a_n)=0$, then $h(a_1,a_2,\dots ,a_n)=0$. As a corollary, it follows that for $f\in {𝕄}_n(S)[x_1,x_2,\dots ,x_n]$, $S$ commutative ring, if $A_1,A_2,\dots ,A_n\in {𝕄}_n(S)$ are pairwise commuting matrices such that $f(A_1,A_2,\dots ,A_n)=0$, then $g(A_1,A_2,\dots ,A_n)=0$ where $g=\mathrm{\text{det}}(f)I$. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer $n>1$, we prove that if every matrix in ${𝕋}_n(R)$ satisfies a monic polynomial of degree $n$ over $R$, then $R$ is commutative. On the other hand, every diagonal matrix in ${𝕄}_n(R)$, $n>1$, satisfies a monic polynomial of degree $n$ over $R$ precisely when $R$ is a left duo ring. We prove that if every diagonal matrix in ${𝕄}_n(R)$, $n>1$, is $R$-integral, then $R$ is Dedekind finite.

In this paper, we introduce the notions of $S_{\varphi }$-polynomial and $S$-minimal value set polynomial where $\varphi$ is a polynomial over a finite field ${𝔽}_q$ and $S$ is a finite subset of an algebraic closure of ${𝔽}_q$. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are $S_{\varphi }$-minimal value set polynomials for $\varphi =x^m$, whose $S$-value sets can be explicitly computed in terms of the monomial $x^m$.

Let $R$ be a ring and let $n$ be an arbitrary but fixed positive integer. We characterize those rings $R$ whose elements $a$ satisfy at least one of the relations that $a^n+a$ or $a^n-a$ is a nilpotent whenever $n\in \mathbb{N}\setminus \{1\}$. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with $x^n-x$ nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with $x^n-x$ nilpotent, J. Algebra Appl. 21 (2022)], respectively.

Let $k$ be a positive integer. For a commutative ring $R$, the ring of dual numbers of $k$ variables over $R$ is the quotient ring $R[x_1,\dots ,x_k]/I$, where $I$ is the ideal generated by the set $\{x_i{x}_j|i,j\in \{1,\dots ,k\}\}$. This ring can be viewed as $R[{\alpha }_1,\dots ,{\alpha }_k]$ with ${\alpha }_i{\alpha }_j=0$, where ${\alpha }_i=x_i+I$ for $1\le i,j\le k$. We investigate the polynomial functions of $R[{\alpha }_1,\dots ,{\alpha }_k]$ whenever $R$ is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on $R[{\alpha }_1,\dots ,{\alpha }_k]$ depending on the order of the pointwise stabilizer of the subring of constants $R$ in the group of polynomial permutations of $R[{\alpha }_1,\dots ,{\alpha }_k]$. Further, we show that the stabilizer group of $R$ is independent of the number of variables $k$. Moreover, we prove that a function $F$ on $R[{\alpha }_1,\dots ,{\alpha }_k]$ is a polynomial function if and only if a system of linear equations on $R$ that depends on $F$ has a solution.

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ₁, …, ξ_t by conjugate algebraic numbers of bounded degree over ℚ, provided that the given transcendental numbers ξ₁, …, ξ_t generate over ℚ a field of transcendence degree one. We provide sharper estimates for example when ξ₁, …, ξ_t form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ₁, …, ξ_t.

In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.

In this paper we investigate the factorization of trinomials of the form xⁿ + cx^n-1 + d ∈ ℤ[x]. We then use these results about trinomials to prove results about the factorization of polynomials of the form xⁿ + c(x^n-1 +⋯+ x + 1) ∈ ℤ[x].

Let S be a finite set of rational primes. For a non-zero integer n, define , where |n|_p is the usual p-adic norm of n. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for [n(n+1)⋯(n+k)]_S for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(n + 1)]_S and [n² + 7]_S, where S = {2, 3} and S = {2}, respectively. We extend Stewart's theorem to prove effective upper bounds for [f(n)]_S for an arbitrary f(x) in ℤ[x] having at least two distinct roots.

For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of components of their functional graphs as well as the average number of periodic points of their associated dynamical systems.