Recent zbMATH articles in MSC 12https://zbmath.org/atom/cc/122022-11-17T18:59:28.764376ZWerkzeugEmbedded Picard-Vessiot extensionshttps://zbmath.org/1496.031572022-11-17T18:59:28.764376Z"Brouette, Quentin"https://zbmath.org/authors/?q=ai:brouette.quentin"Cousins, Greg"https://zbmath.org/authors/?q=ai:cousins.greg"Pillay, Anand"https://zbmath.org/authors/?q=ai:pillay.anand"Point, Francoise"https://zbmath.org/authors/?q=ai:point.francoiseSummary: We prove that if \(T\) is a theory of large, bounded, fields of characteristic 0 with almost quantifier elimination, and \(T_D\) is the model companion of \(T\cup\{\)``\(\partial\) is a derivation''\(\}\), then for any model \((\mathcal U,\partial)\) of \(T_D\), differential subfield \(K\) of \(\mathcal U\) such that \(C_K\vDash T\), and linear differential equation \(\partial Y= AY\) over \(K\), there is a Picard-Vessiot extension \(L\) of \(K\) for the equation with \(K\leq L\leq\mathcal U\), i.e. \(L\) can be embedded in \(\mathcal U\) over \(K\), as a differential field. Moreover such \(L\) is unique to isomorphism over \(K\) as a differential field. Likewise for the analogue for strongly normal extensions for logarithmic differential equations in the sense of Kolchin.Well quasi-orderings and roots of polynomials in a Hahn fieldhttps://zbmath.org/1496.031582022-11-17T18:59:28.764376Z"Knight, Julia F."https://zbmath.org/authors/?q=ai:knight.julia-f"Lange, Karen"https://zbmath.org/authors/?q=ai:lange.karenSummary: Let \(G\) be a divisible ordered abelian group, and let \(K\) be a field. The \textit{Hahn field} \(K((G))\) is a field of formal power series, with terms corresponding to elements in a well ordered subset of \(G\) and the coefficients coming from \(K\). Ideas going back to Newton show that if \(K\) is either algebraically closed of characteristic 0, or real closed, then the same is true for \(K((G))\). Results of \textit{M. H. Mourgues} and \textit{J. P. Ressayre} [J. Symb. Log. 58, No. 2, 641--647 (1993; Zbl 0786.12005)] led us to look for bounds on the lengths of roots of a polynomial, in terms of the lengths of the coefficients [\textit{J. F. Knight} and \textit{K. Lange}, Proc. Lond. Math. Soc. (3) 107, No. 1, 177--197 (2013; Zbl 1294.03029); Sel. Math., New Ser. 25, No. 1, Paper No. 14, 36 p. (2019; Zbl 1428.12011)]. In the present paper, we give an introduction to Hahn fields, we indicate how well quasi-orderings arise when we try to bound the lengths of sums and products, and we re-work, in a more general way, a technical theorem from [loc. cit.] that gives information on the root-taking process.
For the entire collection see [Zbl 1443.03002].Imaginaries and invariant types in existentially closed valued differential fieldshttps://zbmath.org/1496.031592022-11-17T18:59:28.764376Z"Rideau, Silvain"https://zbmath.org/authors/?q=ai:rideau.silvainSummary: We answer three related open questions about the model theory of valued differential fields introduced by \textit{T. Scanlon} [J. Symb. Log. 65, No. 4, 1758--1784 (2000; Zbl 0977.03021)]. We show that they eliminate imaginaries in the geometric language introduced by \textit{D. Haskell} et al. [J. Reine Angew. Math. 597, 175--236 (2006; Zbl 1127.12006)] and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.Magnums: counting sets with surrealshttps://zbmath.org/1496.031822022-11-17T18:59:28.764376Z"Lynch, Peter"https://zbmath.org/authors/?q=ai:lynch.peterSummary: How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This jars with our intuition.
The class of surreal numbers \(\mathbb S\) is the largest possible ordered field. All the basic arithmetical operations are defined, and sensible arithmetic can be carried out over \(\mathbb S\). Using the surreals, we define the `magnum' for subsets of the natural numbers. The magnum of a proper subset of a set is strictly less than the magnum of the set itself.
For the entire collection see [Zbl 1496.00073].Introduction to \(N\)-soft algebraic structureshttps://zbmath.org/1496.032142022-11-17T18:59:28.764376Z"Kamaci, Hüseyin"https://zbmath.org/authors/?q=ai:kamaci.huseyinSummary: This paper is dedicated to two main objectives. The first of these is to develop some new operations on \(N\)-soft set, which is the generalization of soft set. The second is to highlight the concepts of \(N\)-soft group, \(N\)-soft ring, \(N\)-soft ideal, completely semiprime \(N\)-soft ideal, \(N\)-soft field and \(N\)-soft lattice. Moreover, in this study, it is attempted to derive certain properties for these concepts and to analyze the relations between them.Algebraicity modulo \(p\), hypergeometric series and strong Frobenius structurehttps://zbmath.org/1496.111002022-11-17T18:59:28.764376Z"Vargas-Montoya, Daniel"https://zbmath.org/authors/?q=ai:vargas-montoya.danielSummary: This work is devoted to study of algebraicity modulo \(p\) of Siegel's \(G\)-functions. Our goal is emphasize the relevance of the notion of strong Frobenius structure, classically studied in the theory of the \(p\)-adic differential equations, for the study of a Adamczewski-Delaygue's conjecture concerning the degree of algebraicity modulo \(p\) of \(G\)-functions. For this, we first make a Christol's result explicit by showing that if \(f(z)\) is a \(G\)-function which is a solution of a differential operator in \(\mathbb{Q}(z)[d/dz]\) of order \(n\) which has a strong Frobenius structure with period \(h\) for the prime number \(p\) and that \(f(z)\) belongs to \(\mathbb{Z}_{(p)}[[z]]\), then the reduction of \(f(z)\) modulo \(p\) is algebraic over \(\mathbb{F}_p(z)\) and its degree of algebraicity is bounded by \(p^{n^2h} \). By generalizing an approach introduced by Salinier, we then show that a Fuchsian operator with coefficients in \(\mathbb{Q}(z)\), whose monodromy group is rigid and whose exponents are rational has for almost all prime numbers \(p\) a strong Frobenius structure with period \(h\), where \(h\) is explicitly bounded and does not depend on \(p\). A slightly different version of this result has been demonstrated recently by Crew following a different approach based on the \(p\)-adic cohomology. We use these two results to solve the mentioned conjecture in the case of generalized hypergeometric series.Preimages of \(p\)-linearized polynomials over \(\mathbb{F}_p\)https://zbmath.org/1496.111372022-11-17T18:59:28.764376Z"Kim, Kwang Ho"https://zbmath.org/authors/?q=ai:kim.kwang-ho"Mesnager, Sihem"https://zbmath.org/authors/?q=ai:mesnager.sihem"Choe, Jong Hyok"https://zbmath.org/authors/?q=ai:choe.jong-hyok"Lee, Dok Nam"https://zbmath.org/authors/?q=ai:lee.dok-namSummary: Linearized polynomials over finite fields have been intensively studied over the last several decades. Interesting new applications of linearized polynomials to coding theory and finite geometry have been also highlighted in recent years. Let \(p\) be any prime. Recently, preimages of the \(p\)-linearized polynomials \(\sum_{i=0}^{\frac{k}{l}-1} X^{p^{li}}\) and \(\sum_{i=0}^{\frac{k}{l}-1} (-1)^i X^{p^{li}}\) were explicitly computed over \(\mathbb{F}_{p^n}\) for any \(n\). This paper extends that study to \(p\)-linearized polynomials over \(\mathbb{F}_p\), i.e., polynomials of the shape
\[
L(X)=\sum\limits_{i=0}^t \alpha_i X^{p^i}, \alpha_i\in\mathbb{F}_p.
\]
Given a \(k\) such that \(L(X)\) divides \(X-X^{p^k}\), the preimages of \(L(X)\) can be explicitly computed over \(\mathbb{F}_{p^n}\) for any \(n\).The d'Alembert-Gauss theoremhttps://zbmath.org/1496.120012022-11-17T18:59:28.764376Z"Hauchecorne, Bertrand"https://zbmath.org/authors/?q=ai:hauchecorne.bertrandThe fundamental theorem of algebra, which asserts that every polynomial of degree \(n \ge 1\) with real coefficients has at least one (real or complex) root, is known in France as the Theorem of d'Alembert-Gauss. This article briefly sketches the contributions to this result by Bombelli, Girard, d'Alembert, Gauss and Cauchy.
Reviewer: Franz Lemmermeyer (Jagstzell)Polynomials over ring of integers of global fields that have roots modulo every finite indexed subgrouphttps://zbmath.org/1496.120022022-11-17T18:59:28.764376Z"Mishra, Bhawesh"https://zbmath.org/authors/?q=ai:mishra.bhaweshThis paper proves results about intersective polynomials. A polynomial with coefficients in the ring of integers \(O_K\) of a global field \(K\) is called intersective if it has a root modulo every finite-indexed subgroup of \(O_K\). The main result shown by the author gives two equivalent conditions for a polynomial \(f(x)\in O_K[x]\) to be intersective. One of these conditions involves the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon \(K\) and the polynomial \(f\) itself. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.
After the introduction, the rest of the article is organized into three sections. The second section contains statements and proofs of results that are required to prove the main result. Section 3 contains the proof of the main result. Section 4 contains some examples, consequences and discussions pertaining to the theorem.
Reviewer: Mouad Moutaoukil (Fès)A class of fields with a restricted model completeness propertyhttps://zbmath.org/1496.120032022-11-17T18:59:28.764376Z"Dittmann, Philip"https://zbmath.org/authors/?q=ai:dittmann.philip"Leijnse, Dion"https://zbmath.org/authors/?q=ai:leijnse.dionRecall that a first-order theory of fields is called model-complete if, modulo its axioms, every formula is logically equivalent to an existential formula. Or equivalently, every definable set is the projection of a constructible set.
The main result of the paper is the following, which can be viewed as a kind of restricted model-completeness property.
\textbf{Theorem.} The following are equivalent for a field \(K\).
\begin{itemize}
\item[(1)] For every pair of elementary extensions \(K^{\ast\ast}\), \(K^\ast\) of \(K\) with \(K^{\ast \ast}\supseteq K^\ast\), the extension \(K^{\ast\ast}/K^\ast\) is a regular extension of fields.
\item[(2)] For every 3-tuple \((m, d, r)\) there is an existential criterion for the ideal \((f_1, \ldots, f_r)\subseteq K[X_1, \ldots, X_m]\) generated by polynomials \(f_i\) of total degree at most \(d\) to be prime.
\item[(3)] For every 3-tuple \((m, d, r)\) there is an existential criterion for the ideal \((f_1, \ldots, f_r)\subseteq K[X_1, \ldots, X_m]\) generated by polynomials \(f_i\) of total degree at most \(d\) to be maximal.
\end{itemize}
These statements imply the following, and are equivalent to it if \(K\) has finite degree of imperfection, i.e., if the extension \(K^{1/p}/K\) is finite, where \(p\) is the characteristic exponent of \(K\).
\begin{itemize}
\item[(4)] For every quantifier-free formula \(\psi(x, \bar y)\) in the language of rings where \(x\) is a distnguished variable, the formula \(\forall x\psi(x, \bar y)\)is equivalent over \(K\) to an existential formula with parameters in \(K\).
\end{itemize}
Previous results from the first author imply that global fields satisfy condition (1) : (from the Introduction) ``As results from [\textit{P. Dittmann}, Compos. Math. 154, No. 4, 761--772 (2018; Zbl 1429.11220)] imply that global fields satisfy (1), the main theorem in particular implies that global fields satisfy (2), (3), and (4), so we gain new existential definitions in global fields (Corollary 3.2).''.
Reviewer: Luc Bélair (Montréal)On degree of nonlinearity of the coordinate polynomials for a product of transformations of a binary vector spacehttps://zbmath.org/1496.150242022-11-17T18:59:28.764376Z"Fomichëv, Vladimir Mikhaĭlovich"https://zbmath.org/authors/?q=ai:fomichev.vladimir-mikhailovichSummary: We construct a nonnegative integer matrix to evaluate the matrix of nonlinearity characteristics for the coordinate polynomials of a product of transformations of a binary vector space. The matrix of the characteristics of the transformation is defined by the degrees of nonlinearity of the derivatives of all coordinate functions with respect to each input variable. The entries of the evaluation matrix are expressed in terms of the characteristics of the coordinate polynomials of the multiplied transformations. Calculation of the evaluation matrix is easier than calculating the exact values of the characteristics. The estimation method is extended to an arbitrary number of multiplied transformations. Computational examples are given that in particular show the accuracy of the obtained estimates and the domain of their nontriviality.Universal enveloping of (modified) \(\lambda\)-differential Lie algebrashttps://zbmath.org/1496.160252022-11-17T18:59:28.764376Z"Peng, Xiao-Song"https://zbmath.org/authors/?q=ai:peng.xiao-song"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.3|zhang.yi.2|zhang.yi.1|zhang.yi.6|zhang.yi.12|zhang.yi.14|zhang.yi.8|zhang.yi.5|zhang.yi.10|zhang.yi.4|zhang.yi"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Luo, Yan-Feng"https://zbmath.org/authors/?q=ai:luo.yan-fengThis paper deals with somewhat natural generalizations of differential (associative or Lie) algebras, namely \(\lambda\)-differential (associative or Lie) algebras and modified \(\lambda\)-differential (associative or Lie) algebras. A \(\lambda\)-differential (associative or Lie) algebra, for a constant \(\lambda\), is roughly an associative or Lie algebra \(A\) with a linear endomorphism \(A\xrightarrow{d}A\) which satisfies a relation similar to the Leibniz rule, namely \(d(xy)=xd(y)+d(x)y+\lambda d(x)d(y)\), \(x,y\in A\) (where, when \(A\) is a Lie algebra, a concatenation such as \(xy\) should be read as a Lie bracket \([x,y]\)). In the ``modified'' version the term \(\lambda d(x)d(y)\) is replaced by the term \(\lambda xy\).
The existence and uniqueness (up to a unique isomorphism) results about free objects are obtained easily by the authors because \(\lambda\)-differential \((\Bbbk,\partial)\)-modules and algebras, and their ``modified'' versions are categories concretely equivalent to varieties of algebras in the sense of universal algebra so that each algebraic functor, that is, a functor which preserves the underlying sets, has a left adjoint. Therefore the authors focus on explicit constructions for these objects, which is far more interesting and more tricky (see, e.g., Theorem 3.5, p.~1114, Theorem~3.8, p.~1116).
Reviewer: Laurent Poinsot (Villetaneuse)On Landesman-Lazer conditions and the fundamental theorem of algebrahttps://zbmath.org/1496.340752022-11-17T18:59:28.764376Z"Amster, Pablo"https://zbmath.org/authors/?q=ai:amster.pabloIn this paper, the author deals with the differential system
\[
u'(t)+g(u(t))=p(t), \tag{1}
\]
where \(g:\mathbb{R}^2 \rightarrow \mathbb{R}^2\) is bounded and \(p\) is continuous and \(T\)-periodic. Two results for the existence of at least one \(T\)-periodic solution for system (1) are obtained when \(g\) satisfies Landesman-Lazer type conditions. The connection of the second result with the fundamental theorem of algebra is stated.
Furthermore, the author treats the following delay systems
\[
u'(t)=g(u(t))+p(t,u(t),u(t-\tau)), \tag{2}
\]
where \(\tau>0\) and \(p\) is bounded, continuous and \(T\)-periodic in the first coordinate. Under similar conditions, two theorems for the existence of at least one \(T\)-periodic solution for system (2) are proved.
Reviewer: Chun Li (Chongqing)Hyperstability of orthogonally 3-Lie homomorphism: an orthogonally fixed point approachhttps://zbmath.org/1496.460782022-11-17T18:59:28.764376Z"Keshavarz, Vahid"https://zbmath.org/authors/?q=ai:keshavarz.vahid"Jahedi, Sedigheh"https://zbmath.org/authors/?q=ai:jahedi.sedighehSummary: In this chapter, by using the orthogonally fixed point method, we prove the Hyers-Ulam stability and the hyperstability of orthogonally 3-Lie homomorphisms for additive \(\rho\)-functional equation in 3-Lie algebras. Indeed, we investigate the stability and the hyperstability of the system of functional equations
\[
\begin{cases}
f(x+y)-f(x)-f(y)= \rho (2f(\frac{x+y}{2})+ f(x)+ f(y)),\\
f([[u,v],w])=[[f(u),f(v)],f(w)]
\end{cases}
\]
in 3-Lie algebras where \(\rho \neq 1\) is a fixed real number.
For the entire collection see [Zbl 1485.65002].Polynomial birth-death processes and the 2nd conjecture of Valenthttps://zbmath.org/1496.470572022-11-17T18:59:28.764376Z"Bochkov, Ivan"https://zbmath.org/authors/?q=ai:bochkov.ivanSummary: The conjecture of \textit{G. Valent} [ISNM, Int. Ser. Numer. Math. 131, 227--237 (1999; Zbl 0935.30025)] about the type of Jacobi matrices with polynomially growing weights is proved.Testing polynomial equivalence by scaling matriceshttps://zbmath.org/1496.683772022-11-17T18:59:28.764376Z"Bläser, Markus"https://zbmath.org/authors/?q=ai:blaser.markus"Rao, B. V. Raghavendra"https://zbmath.org/authors/?q=ai:raghavendra-rao.b-v"Sarma, Jayalal"https://zbmath.org/authors/?q=ai:sarma-m-n.jayalalSummary: In this paper we study the polynomial equivalence problem: test if two given polynomials \(f\) and \(g\) are equivalent under a non-singular linear transformation of variables.
We begin by showing that the more general problem of testing whether \(f\) can be obtained from \(g\) by an arbitrary (not necessarily invertible) linear transformation of the variables is equivalent to the existential theory over the reals. This strengthens an \(\mathsf {NP}\)-hardness result by
\textit{N. Kayal} [in: Proceedings of the 44th annual ACM symposium on theory of computing, STOC'12. New York, NY: Association for Computing Machinery (ACM). 643--662 (2012; Zbl 1286.68197)].
Two \(n\)-variate polynomials \(f\) and \(g\) are said to be equivalent up to scaling if there are scalars \(a_1,\ldots,a_n\in\mathbb {F}\setminus\{0\}\) such that \(f(a_1x_1,\ldots,a_nx_n)=g(x_1,\ldots ,x_n)\). Testing whether two polynomials are equivalent by scaling matrices is a special case of the polynomial equivalence problem and is harder than the polynomial identity testing problem.
As our main result, we obtain a randomized polynomial time algorithm for testing if two polynomials are equivalent up to a scaling of variables with black-box access to polynomials \(f\) and \(g\) over the real numbers.
An essential ingredient to our algorithm is a randomized polynomial time algorithm that given a polynomial as a black box obtains coefficients and degree vectors of a maximal set of monomials whose degree vectors are linearly independent. This algorithm might be of independent interest. It also works over finite fields, provided their size is large enough to perform polynomial interpolation.
For the entire collection see [Zbl 1369.68029].