Affine Plane Research Articles

Some geometric structures related with Desargues configuration in PG(2,5)

In this study, we investigate geometric structures related to Desargues configuration in PG(2,5), using an algorithm implemented in C#. We determine different six complete (11,3)-arcs containing Desargues configuration, a (6,2)-arc not containing Desargues configuration, a complete (16,4)- arc, and an affine plane in PG(2,5).

Sets of mutually orthogoval projective and affine planes

Sets of mutually orthogoval projective and affine planes

On free and plus-one generated curves arising from free curves by addition–deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

Special directions on the finite affine plane

In this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p, where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p.

Semi-classical 2-dimensional Minkowski planes

AbstractSemi-classical geometries have been investigated in the context of 2-dimensional affine planes, projective planes, Möbius planes and Laguerre planes. Here we deal with the case of 2-dimensional Minkowski planes. Semi-classical 2-dimensional Minkowski planes are obtained by pasting together two halves of the classical real Minkowski plane along two circles or parallel classes. By solving some functional equations for the functions that describe the pasting we determine all semi-classical 2-dimensional Minkowski planes. In contrast to the rich variety of other semi-classical planes there are only very few models of such Minkowski planes.

Efficient polar coordinates attack with adaptive activation strategy

In the realm of decision-based attacks, which aim to mislead target models by manipulating output labels, the advent of the polar coordinates attack has marked a significant evolution. Distinguished by its utilization of sampled affine planes with unique geometrical properties, this method has demonstrated notable enhancements in query efficiency. However, existing implementations of the latest polar coordinates attack encounter two major problems: a contradiction in the update criteria and an overemphasis on allocating query resources to angle binary search. To address these issues, the Efficient Polar Coordinates Attack (EPCA) is introduced, a novel approach that resolves these concerns while significantly augmenting query efficiency. Furthermore, a mechanism termed Adaptive Activation Strategy is proposed to control the deployment of a recently presented plug-and-play module named Frequency Binary Search. The integration of these elements further enables EPCA to effectively circumvent the local optimum. Abundant experiments conducted on the ImageNet dataset demonstrate the exceptional improvements achieved by EPCA, establishing it as a state-of-the-art method in decision-based attacks. The simulation codes of this paper are accessible at https://github.com/RYC-98/EPCA.

A lower bound on the third-order nonlinearity of the simplest [formula omitted] bent functions

Boolean functions used in symmetric-key encryption should have high higher-order nonlinearity to resist several known cryptographic attacks, such as algebraic attacks and low-degree approximation attacks. The higher-order nonlinearity also plays an important role in coding theory and theoretical computer science, since it relates to the covering radius of Reed–Muller codes and the Gowers norm, respectively. It is well-known that bent functions have the highest nonlinearity in an even number of variables and thus they possess the best ability to withstand fast correlation attacks and best affine approximation attacks. However, there is currently limited knowledge regarding the higher-order nonlinearity of bent functions because computing the higher-order nonlinearity, or even providing tight lower bounds, is an extremely hard task. In 1974, Dillon proposed two well-known classes of bent functions based on partial spread (in brief, PS), called PS− and PS+, respectively. He also exhibited a subclass of bent functions in PS−, known as partial spread affine plane(PSap for short). In this paper, we provide a lower bound on the third-order nonlinearity of the simplest PSap bent functions in n variables, where n≥6 is even, by calculating the nonlinearities of all second-order derivatives of this kind of bent functions. Compared to the two known lower bounds on the third-order nonlinearity given by Carlet and Tang et al. respectively, our lower bound is much better than these two ones.

Embedded complex curves in the affine plane

This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^2$$\\end{document} satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^2$$\\end{document}, and every connected domain in C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^2$$\\end{document} admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of M˚=M\\bM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathring{M}=M\\setminus bM$$\\end{document}, and compact subset K⊂M˚\\E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\subset \\mathring{M}\\setminus E$$\\end{document} without holes in M˚\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathring{M}$$\\end{document}, any C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}^1$$\\end{document} embedding f:M↪C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:M\\hookrightarrow \\mathbb {C}^2$$\\end{document} which is holomorphic in M˚\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathring{M}$$\\end{document} can be approximated uniformly on K by holomorphic embeddings F:M↪C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F:M\\hookrightarrow \\mathbb {C}^2$$\\end{document} which map E∪bM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E\\cup bM$$\\end{document} out of a given ball and satisfy some interpolation conditions.

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Semiaffine stable planes

AbstractA locally compact stable plane of positive topological dimension will be called semiaffine if for every line L and every point p not in L there is at most one line passing through p and disjoint from L. We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general.

A quasidouble of the affine plane of order 4 and the solution of a problem on additive designs

A 2-(v,k,λ) block design (P,B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G,+) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G, the embedding of P in G is also such that, conversely, any zero-sum k-subset of P is a block in B, then (P,B) is said to be strongly additive.In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2-(16,4,2) design (F4×F4,B2), which decomposes into two disjoint isomorphic copies of the affine plane of order four.An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG(4,2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem.

Expressive curves

We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C C is expressive if (a) each irreducible component of C C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.

Renitent lines

There are many examples for point sets in finite geometry which behave “almost regularly” in some (well-defined) sense, for instance they have “almost regular” line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod p, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behavior of these renitent lines. As a consequence of our results, we also prove geometric properties of codewords of the Fp-linear code generated by characteristic vectors of lines of PG(2,q).

Ising model on the affine plane

Ising model on the affine plane

Smooth Affine $\mathbb{G}_m$-surfaces with Finite Picard Groups and Trivial Units

We determine the smooth affine $\mathbb{G}_m$-surfaces of logarithmic Kodaira dimension $\leq 0$ with finite Picard groups and trivial units defined over an algebraically closed field. Furthermore, we prove that a smooth factorial affine surface with trivial units is isomorphic to the affine plane if and only if it admits an effective $\mathbb{G}_m$-action.

Poincaré–Reeb graphs of real algebraic domains

An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincaré–Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincaré–Reeb graph.

A new discrete theory of pseudoconvexity

Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.

Quasi-monomials with respect to subgroups of the plane affine group

Let $H$ be a subgroup of the plane affine group ${\rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family $\{ B_{m,n}(x,y) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ \{ x^m y^n \} $ and $\{ B_{m,n}(x,y) \}$ have \textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family $\{ B_{m,n}(x,y) \}$.

On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties

Abstract Our main result is the following: let $X$ be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface $X^{\prime}$ with automorphism group isomorphic to the automorphism group of $X$ if and only if $X$ is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety $X$ of any dimension. Recall that normalized additive group actions on $X$ are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of $X$. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.