Problems In Algebraic Geometry Research Articles

Solving systems of algebraic equations over finite commutative rings and applications

AbstractSeveral problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

Miyaoka-Yau type inequalities of complete intersection threefolds in products of projective

Geography of projective varieties is one of the fundamental problems in algebraic geometry. There are many researches toward the characteristics of Chern number of some projective spaces, for example Noethers inequalities, the theorem of Chang-Lopez, and the Miyaoka-Yau inequality. In this paper, we compute the Chern numbers of any smooth complete intersection threefold in the product of projective spaces via the standard exact sequences of cotangent bundles. Then we obtain linear Chern number inequalities for (c_1 (X)c_2 (X))/(c_1^3 (X)) and (c_3 (X))/(c_1^3 (X)) on such threefolds under conditions of d_ij4 and d_ij6 respectively. They can be considered as a generalization of the Miyaoka-Yau inequality and an improvement of Yaus inequality for such threefolds.

A note on Bridgeland stability conditions and Catalan numbers

In this short note, we describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface, and the action of its autoequivalence group on its Bridgeland stability manifold. In solving an equation involving this group action, the generating function of Catalan numbers arises, allowing us to use asymptotic estimates of Catalan numbers to arrive at a bound for the solution set.

Learning algebraic structures: Preliminary investigations

In this paper, we employ techniques of machine learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether artificial intelligence (AI) can “learn” algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by “looking” at the Cayley table or correctly matching addition and multiplication tables for finite rings can, at least for structures of small size, be performed by the AI, even after having been trained only on small number of cases. These results are in tandem with recent investigations on whether AI can solve certain classes of problems in algebraic geometry.

Operadic approach to wall-crossing

Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather complicated transformation rules and non-trivial combinatorial problems. In this paper we propose a framework, reminiscent of collections and plethysms in the theory of operads, that conceptualizes those transformation rules. As an application we obtain new streamlined proofs of some existing wall-crossing formulas as well as some new formulas related to attractor invariants.

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g \geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $\overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $\neq 2$. We show that all $F$-forms of $\overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 \leqslant n \leqslant 4$, $g = 2$ and $2 \leqslant n \leqslant 3$, $g = 3$ and $1 \leqslant n \leqslant 14$, $g = 4$ and $1 \leqslant n \leqslant 9$, $g = 5$ and $1 \leqslant n \leqslant 12$.

Resolvent degree, Hilbert’s 13th Problem and geometry

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.

The CR Immersion into a Sphere with the Degenerate CR Gauss Map

It is a classical problem in algebraic geometry to characterize the algebraic subvariety by using the Gauss map. In this note, we try to develop the analogue theory in CR geometry. In particular, under some assumptions, we show that a CR map between spheres is totally geodesic if and only if the CR Gauss map of the image is degenerate.

Machine learning CICY threefolds

The latest techniques from Neural Networks and Support Vector Machines (SVM) are used to investigate geometric properties of Complete Intersection Calabi–Yau (CICY) threefolds, a class of manifolds that facilitate string model building. An advanced neural network classifier and SVM are employed to (1) learn Hodge numbers and report a remarkable improvement over previous efforts, (2) query for favourability, and (3) predict discrete symmetries, a highly imbalanced problem to which both Synthetic Minority Oversampling Technique (SMOTE) and permutations of the CICY matrix are used to decrease the class imbalance and improve performance. In each case study, we employ a genetic algorithm to optimise the hyperparameters of the neural network. We demonstrate that our approach provides quick diagnostic tools capable of shortlisting quasi-realistic string models based on compactification over smooth CICYs and further supports the paradigm that classes of problems in algebraic geometry can be machine learned.

Lifting of polynomial symplectomorphisms and deformation quantization

ABSTRACTWe study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory.This paper is a continuation of the study [19].

THE ANALYSIS OF IMAGES IN N-POINT GRAVITATIONAL LENS BY METHODS OF ALGEBRAIC GEOMETRY

<p>This paper is devoted to the study of images in <em>N</em>-point gravitational lenses by methods of algebraic geometry. In the beginning, we carefully define images in algebraic terms. Based on the definition, we show that in this model of gravitational lenses (for a point source), the dimensions of the images can be only 0 and 1. We reduce it to the fundamental problem of classical algebraic geometry - the study of solutions of a polynomial system of equations. Further, we use well-known concepts and theorems. We adapt known or prove new assertions. Sometimes, these statements have a fairly general form and can be applied to other problems of algebraic geometry. In this paper: the criterion for irreducibility of polynomials in several variables over the field of complex numbers is effectively used. In this paper, an algebraic version of the Bezout theorem and some other statements are formulated and proved. We have applied the theorems proved by us to study the imaging of dimensions 1 and 0.</p>

Algebraic K-theory and Motivic Cohomology

Algebraic $K$-theory and motivic cohomology have developed together over the last thirty years. Both of these theories rely on a mix of algebraic geometry and homotopy theory for their construction and development, and both have had particularly fruitful applications to problems of algebraic geometry, number theory and quadratic forms. The homotopy-theory aspect has been expanded significantly in recent years with the development of motivic homotopy theory and triangulated categories of motives, and $K$-theory has provided a guiding light for the development of non-homotopy invariant theories. 19 one-hour talks presented a wide range of latest results on many aspects of the theory and its applications.

Формальное представление условий инцидентности в многомерных проективных пространствах

Synthetic and analytical methods are usually used to investigate multidimensional spaces and sets of subspaces. Shortcomings of a synthetic method — the appeal to spatial imagination and the researcher´s intuition, impossibility of formalization, need of creation of big and complex logical constructions don´t allow to go beyond four-dimensional space, with rare exceptions. The theory of enumerative geometry submitted as geometry of conditions with a basic element – a multidimensional flags incidence condition allows solve many problems which up to now were considered as insoluble ones. The simplest of such problems is a classical problem for calculation of final number of given space’s subspaces meeting the set of given conditions (the normal problem of algebraic geometry). A more serious problem – calculation a number and values of algebraic characteristics for given variety in given space. For this problem solution it was necessary to develop a formalized method, as well as technique algorithmization. This problem has been solved by professor. V.Ya. Volkov in his doctoral dissertation by means of developed by him so-called e-calculations. For an understanding of e-calculation fundamentals or calculation of Schubert conditions a rather good mathematical background and method promotion are necessary. The last demands consideration of various approaches to conditions calculation problem. In the present paper are considered the simplest cases of a tabular method for conditions calculation, in relation to conditions of incidence which is understood in a general sense. Calculation of one-, two- and ((k + 1) (n – k) – 1)-dimensional conditions are considered. Conditions calculation formalization and incidence conditions reduction are explained. The problem about a final number of straight lines crossing the set number of k-planes in n-dimensional space is considered as an example. In particular, the problem about a number of straight lines crossing some number of the set straight lines can be correct only in three- and four-dimensional spaces. Conditions of the minimum multiplicity equal to three exist only in (3k + 1)-dimensional spaces. Conditions of the multiplicity equal to four exist only in odd dimensionality spaces. And so on. The concrete number of straight lines in all cases can be counted by reduction of the corresponding conditions.

Solving Geometric Problems by Using Algebraic Representation for Junior High School Level 3 in Van Hiele at Geometric Thinking Level

<p class="apa">This study aims to determine the ability of algebra students who have 3 levels van Hiele levels. Follow its framework Dindyal framework (2007). Students are required to do 10 algebra shaped multiple choice, then students work 15 about the geometry of the van Hiele level in the form of multiple choice questions. The question has been tested levels of validity and reliability. After learning abilities and levels van Hiele algebra, students were asked to answer two questions descriptions to determine the ability of students in answering the question of algebraic geometry punctuated by interviews. From this study illustrated that students who have achieved level 3 van Hiele able to properly solve problems of algebraic geometry in the content by utilizing the deduction reasoning thinking skills to build the structure geometry in an axiomatic system in solving the problems faced. Teachers play an important role in pushing the speed students through a higher level of thinking through the right exercises. Suggestions for further research can develop on different topics but still within the context of algebraic geometry.</p>

Geometric complexity theory V: Efficient algorithms for Noether normalization

We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether’s Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show the following: (1) The categorical quotient for any finite dimensional representation V V of S L m SL_m , with constant m m , is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of V V . (3) The categorical quotient of the space of r r -tuples of m × m m \times m matrices by the simultaneous conjugation action of S L m SL_m is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in m m and r r in any characteristic p ∉ [ 2 , ⌊ m / 2 ⌋ ] p \not \in [2,\lfloor m/2 \rfloor ] . (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

On asymptotics and resurgent structures of enumerative Gromov–Witten invariants

Making use of large-order techniques in asymptotics and resurgent analysis, this work addresses the growth of enumerative Gromov–Witten invariants—in their dependence upon genus and degree of the embedded curve—for several different threefold Calabi–Yau toric-varieties. In particular, while the leading asymptotics of these invariants at large genus or at large degree is exponential, at combined large genus and degree it turns out to be factorial. This factorial growth has a resurgent nature, originating via mirror symmetry from the resurgent-transseries description of the B-model free energy. This implies the existence of nonperturbative sectors controlling the asymptotics of the Gromov–Witten invariants, which could themselves have an enumerative-geometry interpretation. The examples addressed include: the resolved conifold; the local surfaces local P2 and local P1 × P1; the local curves and Hurwitz theory; and the compact quintic. All examples suggest very rich interplays between resurgent asymptotics and enumerative problems in algebraic geometry.

On undulation invariants of plane curves

One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has singularities or other distinctive features of interest). A classical example of such a problem, described by A.Cayley and G.Salmon in 1852, is to determine whether or not a given plane curve of degree r > 3 has undulation points -- the points where the tangent line meets the curve with multiplicity four. They proved that there exists an invariant of degree 6(r - 3)(3 r - 2) that vanishes if and only if the curve has undulation points. In this paper we give explicit formulae for this invariant in the case of quartics (r=4) and quintics (r=5), expressing it as the determinant of a matrix with polynomial entries, of sizes 21 times 21 and 36 times 36 respectively.

Algebraic Dynamical Systems and Dirichlet's Unit Theorem on Arithmetic Varieties

The effectivity of pseudo-effective |$\mathbb R$|-divisor is an important and difficult problem in algebraic geometry. In the Arakelov geometry framework, this problem can be considered as a generalization of Dirichlet's unit theorem for number fields. In this article, we propose obstructions to the Dirichlet property by two approaches, that is, the denseness of nonpositive points and functionals on adelic |$\mathbb R$|-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic |${\mathbb {R}}$|-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of the Dirichlet property.

On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k) such that (m+1) generic skew-symmetric (k+1)-forms in (n+1) variables as linear combinations of the same s(m,n,k) decomposable skew-symmetric (k+1)-forms.This problem is analogous to a well known problem called Waringʼs problem for symmetric forms and can be very naturally translated into a classical problem in algebraic geometry. In this paper, we will go through some basics of algebraic geometry, describe how objects in algebraic geometry can be associated to systems of skew-symmetric forms, and discuss algebro-geometric approaches to establish the existence of triples (m,n,k), where s(m,n,k) is more than expected.

Determinantal representations and the Hermite matrix

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Hermite matrix. If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares. Conversely, we show how a determinantal representation can sometimes be constructed from a sums-of-squares decomposition of the Hermite matrix. We finally show that definite determinantal representations always exist, if one allows for denominators.