Recent zbMATH articles in MSC 14https://zbmath.org/atom/cc/142021-01-08T12:24:00+00:00WerkzeugThe adaptive ``window'' method for elliptic curve operations.https://zbmath.org/1449.140072021-01-08T12:24:00+00:00"Spel'nikov, A. B."https://zbmath.org/authors/?q=ai:spelnikov.a-bSummary: The algorithm of adaptive ``window'' multiplication of a point by a number on the elliptic curve is introduced, allowing to increase the calculation speed in cryptographic applications.ACC conjecture for weighted log canonical thresholds.https://zbmath.org/1449.140022021-01-08T12:24:00+00:00"Hong, N. X."https://zbmath.org/authors/?q=ai:hong.nguyen-xuan.1"Long, T. V."https://zbmath.org/authors/?q=ai:long.tang-van"Trang, P. N. T."https://zbmath.org/authors/?q=ai:trang.pham-nguyen-thuFor \(n\geq 1\) and \(\mu\) a Borel measure in \(\mathbb{C}^n\), the weighted log canonical threshold of a holomorphic function \(f\) defined in a neighborhood of the origin of \(\mathbb{C}^n\) is defined as \(c_{\mu}(f):=\)sup\(\{ c\geq 0\) : \(|f|^{-2c}\) is \(L^1(\mu)\) in a neighbourhood of \(0~\}\). Set
\(\mathcal{C}(\mu):=\{ c_{\mu}(f)\) : \(f\) is holomorphic in a neighbourhood of \(0~\}\).
The ACC conjecture for weight \(\mu\) reads:
\(\mathcal{C}(\mu)\) satisfies the ascending chain condition every convergent increasing sequence in \(\mathcal{C}(\mu)\) to be stationary.
The authors show that the conjecture holds true for \(n=2\) and \(\mu =\| z\| ^{2t}dV_4\), \(t\geq 0\).
Reviewer: Vladimir P. Kostov (Nice)Complete addition formulas on the level four theta model of elliptic curves.https://zbmath.org/1449.140062021-01-08T12:24:00+00:00"Fouotsa, Emmanuel"https://zbmath.org/authors/?q=ai:fouotsa.emmanuel"Diao, Oumar"https://zbmath.org/authors/?q=ai:diao.oumarSummary: The addition formula provided for the recently discovered level four theta model of elliptic curve [the authors, ibid. 26, No. 3--4, 283--301 (2015; Zbl 1327.14155)] is not complete; meaning that it does not work for all inputs. In this work we provide three alternative addition formulas to solve this problem. The obtained formulas are unified in the sense that they can be used for both addition of points and doubling. These formulas are also valid both in odd and even characteristics. In particular our new formula in binary fields improves the one previously obtained on this curve [loc. cit.]. We compute the cost of each formula. We also provide a sage and magma codes (Fouotsa and Diao in Sage and magma code for the verification of addition formulas, algorithm for cost computation and completeness of addition formulas. \url{http://www.emmanuelfouotsa-prmais.org/Portals/22/CODE.txt}, 2018) to ensure the correctness of all formulas and algorithms in this work.Elliptic threshold secret division scheme.https://zbmath.org/1449.140082021-01-08T12:24:00+00:00"Spel'nikov, A. B."https://zbmath.org/authors/?q=ai:spelnikov.a-bSummary: The new aspect of proactive systems realization was developed based on elliptic curves arithmetic. Neural network model of secret division scheme at elliptic curve is introduced. Prolongation mechanism of the scheme was developed. Neural network model of secret regeneration is introduced. Method of construction and functioning report of the scheme were developed. Calculation of safe time period of secret generator settings existence is introduced. Calculation of complexity and time needed for prolongation of this model is introduced.Testing Lipschitz non-normally embedded complex spaces.https://zbmath.org/1449.140012021-01-08T12:24:00+00:00"Denkowski, Maciej"https://zbmath.org/authors/?q=ai:denkowski.maciej-p"Tibăr, Mihai"https://zbmath.org/authors/?q=ai:tibar.mihai-mariusOne can define two metrics on a locally closed path-connected set \(X\subset \mathbb{R}^n\): the Euclidean metric \(d(x,y)\) and the inner metric \(d_X(x,y)\), where \(d_x,y)\) is the length of the shortest path belonging to \(X\) and joining the points \(x,y\in X\). The set \(X\) is said to be (Lipschitz) normally embedded if there exists \(C>0\) such that \(d_X(x,y)\leq Cd(x,y)\) for all \(x,y\in X\). The authors introduce a sectional criterion for testing if complex analytic germs \((X,0)\subset (\mathbb{C}^n,0)\) are Lipschitz non-normally embedded.
Reviewer: Vladimir P. Kostov (Nice)Tangent space of algebraic manifold in singularity.https://zbmath.org/1449.140032021-01-08T12:24:00+00:00"Hu, Shuangnian"https://zbmath.org/authors/?q=ai:hu.shuangnian"Li, Yanyan"https://zbmath.org/authors/?q=ai:li.yanyan.1|li.yanyan"Qin, Zhentao"https://zbmath.org/authors/?q=ai:qin.zhentao"Niu, Yujun"https://zbmath.org/authors/?q=ai:niu.yujunSummary: In this paper, by using a new tool of limit equation, we study the tangent space of singular points of algebraic manifolds from the point of elementary calculus. Then we get that the tangent space of singular points of general algebraic manifolds is determined by defining the local part of the algebraic manifold system.Split quaternions and time-like constant slope surfaces in Minkowski 3-space.https://zbmath.org/1449.140102021-01-08T12:24:00+00:00"Babaarslan, Murat"https://zbmath.org/authors/?q=ai:babaarslan.murat"Yayli, Yusuf"https://zbmath.org/authors/?q=ai:yayli.yusufSummary: In the present paper, we prove that time-like constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit time-like split quaternions and also homothetic motions. Afterwards we give some examples to illustrate our main results by using Mathematica.Rational solutions of the Diophantine equations \(f(x)^2\pm f(y)^2=z^2\).https://zbmath.org/1449.110622021-01-08T12:24:00+00:00"Youmbai, Ahmed El Amine"https://zbmath.org/authors/?q=ai:youmbai.ahmed-el-amine"Behloul, Djilali"https://zbmath.org/authors/?q=ai:behloul.djilaliIn the paper under review the authors show that for any \(n\ge 1\), there are polynomials \(f(x)\in {\mathbb{Z}}[x]\) of degree \(n\) without multiple roots such that the title equation has infinitely many non-trivial rational solutions \((x,y,z)\). For the proof, they construct some particular polynomials which for certain evaluations of \(x\) and \(y\) in terms of a third parameter \(T\), the resulting equation in \((z,T)\) yields an elliptic curve with a particular rational point on it which is not torsion; hence, any elliptic curve multiple of it yields a distinct solution to the desired equation. Some of the calculations were carried out with Magma.
Reviewer: Florian Luca (Johannesburg)Note on the Steinhaus problem.https://zbmath.org/1449.110752021-01-08T12:24:00+00:00"Guan, Xungui"https://zbmath.org/authors/?q=ai:guan.xunguiSummary: In the paper, by pointing out some errors in other articles, we knew that the well known integer distance point problem of Steinhaus was still open. Using the property of Pythagoras triple and the infinite descent method, some statements concerning the nonexistence of Steinhaus points were obtained, and an open problem given by other articles was partly solved.Rationality and specialization.https://zbmath.org/1449.140052021-01-08T12:24:00+00:00"Tschinkel, Yuri"https://zbmath.org/authors/?q=ai:tschinkel.yuriSummary: I discuss recent advances in the study of rationality properties of algebraic varieties, with an emphasis on the specialization method initiated by Voisin.On the non-existence of negative weight derivations of the new moduli algebras of singularities.https://zbmath.org/1449.140042021-01-08T12:24:00+00:00"Ma, Guorui"https://zbmath.org/authors/?q=ai:ma.guorui"Yau, Stephen S.-T."https://zbmath.org/authors/?q=ai:yau.stephen-shing-toung"Zuo, Huaiqing"https://zbmath.org/authors/?q=ai:zuo.huaiqingTo any isolated hypersurface singularity \((V,0)\subset (\mathbb{C}^{n+1},0),\) defined by a holomorphic function \(f:(\mathbb{C}^{n+1},0)\rightarrow (\mathbb{C},0),\) there are associated many algebraic objects, for instance the Tjurina algebra \(A(V):=\mathcal{O}_{n+1}/(f,\frac{\partial f}{\partial z_{0}},\ldots ,\frac{\partial f}{\partial z_{n+1}})\) which completely determines the analytic structure of \((V,0).\) A weaker invariant is the Lie algebra of derivations of \(A(V)\) i.e. \(L(V):=\operatorname{Der}(A(V),A(V)).\) The authors consider more subtle invariants: the local algebra
\[
A^{\ast }(V):=\mathcal{O}_{n+1}/(f,\frac{\partial f}{\partial z_{0}},\ldots ,\frac{\partial f}{\partial z_{n+1}},\det \left( \frac{\partial ^{2}f}{\partial z_{i}\partial z_{j}}\right) _{0\leq i,j\leq n+1})
\]
and the Lie algebra \(L^{\ast }(V):=\operatorname{Der}(A^{\ast }(V),A^{\ast }(V)).\) They prove the following property of \(L^{\ast }(V)\) for \(1\leq n\leq 3:\) Let \((V,0)\) be an isolated weighted homogeneous hypersurface singularity (then \(f
\) is a polynomial) of a weight type \((\alpha _{0},\ldots ,\alpha _{n+1};d)\) where \(d\geq 2\alpha _{0}\geq \cdots \geq 2\alpha _{n+1}>0.\) (In this case \(A^{\ast }(V)\) and \(L^{\ast }(V)\) are graded algebra). Then in \(L^{\ast }(V)\)
there are no derivations of negative weight i.e. derivations \(D:A^{\ast}(V)\rightarrow A^{\ast }(V)\) which sends elements of degree \(i\) into elements of degree \(i-k_{0},\) where \(k_{0}\in \mathbb{N}.\)
Reviewer's remark: In the paper there is a strange statement ``\dots the second author discovered independently the following conjecture\ldots'' on line~1 on page~202.
Reviewer: Tadeusz Krasiński (Łódź)Generalized algebraic completely integrable systems.https://zbmath.org/1449.700142021-01-08T12:24:00+00:00"Lesfari, Ahmed"https://zbmath.org/authors/?q=ai:lesfari.ahmedSummary: We tackle in this paper the study of generalized algebraic completely integrable systems. Some interesting cases of integrable systems appear as coverings of algebraic completely integrable systems. The manifolds invariant by the complex flows are coverings of abelian varieties and these systems are called algebraic completely integrable in the generalized sense. The later are completely integrable in the sense of Arnold-Liouville. We shall see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense. A large class of algebraic completely integrable systems in the generalized sense, are part of new algebraic completely integrable systems. We discuss some interesting and well known examples: a 4-dimensional algebraically integrable system in the generalized sense as part of a 5-dimensiunal algebraically integrable system, the Hénon-Heiles and a 5-dimensional system, the RDG potential and a 5-dimensional system, the Goryachev-Chaplygin top and a 7-dimensional system, the Lagrange top, the (generalized) Yang-Mills system and cyclic covering of abelian varieties.Higher algebraic \(K\)-theory and representations of algebraic groups.https://zbmath.org/1449.190022021-01-08T12:24:00+00:00"Kuku, Aderemi"https://zbmath.org/authors/?q=ai:kuku.aderemi-oSummary: This paper is concerned with Higher Algebraic \(K\)-theory and actions of algebraic groups \(G\) on such `nice' categories as the category of algebraic vector bundles on a scheme \(X\). Such `nice' categories are examples of `exact' categories with the observation that the category of actions on \(G\) on such exact categories also form an exact category called equivariant exact categories on which one can do higher Algebraic \(K\)-theory (of Quillen) called equivariant higher Algebraic \(K\)-theory -- the higher dimensional generalizations of classical equivariant \(K\)-theory which belongs to the field of representation theory. Thus, for an Algebraic group \(G\) over a number field or \(p\)-adic field \(F\), we present constructions and computations of equivariant higher \(K\)-groups as well as `profinite' or `continuous' higher \(K\)-groups for some \(G\)-Scheme \(X\). In particular, we present explicit l-completeness (l a rational prime) and finiteness computations for higher \(K\)-groups and profinite higher \(K\)-groups for twisted flag varieties.Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one.https://zbmath.org/1449.140092021-01-08T12:24:00+00:00"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.6Let \(X\) be a complex Fano manifold of dimension \(n \geq 3\), and \(A\) be a divisor on \(X\) such that \(A\) is a rational homogeneous space of Picard number \(1\). In [Math. Ann. 342, No. 3, 557--563 (2008; Zbl 1154.14037)] \textit{K. Watanabe} classified the pairs \((X,A)\) when \(A\) is ample. In the paper under review, the author classifies the pairs \((X,A)\) when the conormal bundle \(N_{A|X}^*\) is ample on \(A\). It is shown by \textit{T. Tsukioka} that the Picard number \(\rho(X) \leq 3\) [Geom. Dedicata 123, 179--186 (2006; Zbl 1121.14036)]. To prove the main result of the paper under review, the author describes the extremal contractions of the pairs \((X,A)\) as Casagrande-Druel did for the case when \(\rho(X)=3\) in [\textit{C. Casagrande} and \textit{S. Druel}, Int. Math. Res. Not. 2015, No. 21, 10756--10800 (2015; Zbl 1342.14088)].
Reviewer's remark: There is one minor mistake. It is not true that if \(\mathcal{L}\) is a very ample line bundle, then \(\mathcal{L}\) is simply generated. In Propositions 2.10 and 2.11, the condition that \(\mathcal{L}|_D\) is very ample should be replaced by the condition that \(\mathcal{L}|_D\) is simply generated. These propositions are used when \(D\) is a rational homogeneous space of Picard number \(1\) (in this case \(\mathcal{L}|_D\) is simply generated), so the remaining part of the paper has no trouble.
Reviewer: Jinhyung Park (Seoul)Apollonius ``circle'' in spherical geometry.https://zbmath.org/1449.510232021-01-08T12:24:00+00:00"Ionaşcu, Eugen J."https://zbmath.org/authors/?q=ai:ionascu.eugen-julienSummary: We investigate the analog of the circle of Apollonious in spherical geometry. This can be viewed as the pre-image through the stereographic projection of an algebraic curve of degree three. This curve consists of two connected components each being the ``reflection'' of the other through the center of the sphere. We give an equivalent equation for it, which is surprisingly, this time, of degree four.Rational points of algebraic variety defined by two polynomials.https://zbmath.org/1449.111072021-01-08T12:24:00+00:00"Gao, Wei"https://zbmath.org/authors/?q=ai:gao.wei|gao.wei.4|gao.wei.1|gao.wei.2|gao.wei.3"Huang, Hua"https://zbmath.org/authors/?q=ai:huang.hua"Cao, Wei"https://zbmath.org/authors/?q=ai:cao.weiSummary: Let \({\mathbb{F}_q}\) be the finite field of \(q\) elements. We study rational points on the algebraic variety \(W\) defined by two special polynomials over \({\mathbb{F}_q}\). When the greatest invariant factor of augmented degree matrix of \(W\) is coprime to \(q - 1\), we obtain the explicit formula for the number of \({\mathbb{F}_q}\)-rational points on the algebraic variety \(W\), which generalizes the known results.