Walks on the slit plane: other approaches

Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i belongs to S for all i, and none of the points w_i, i>0, lie on the half-line H= {(k,0): k =< 0}. In a recent paper, G. Schaeffer and the author computed the length generating function S(t) of walks on the slit plane for several sets S. All the generating functions thus obtained turned out to be algebraic: for instance, on the ordinary square lattice, S(t) =\frac{(1+\sqrt{1+4t})^{1/2}(1+\sqrt{1-4t})^{1/2}}{2(1-4t)^{3/4}}. The combinatorial reasons for this algebraicity remain obscure. In this paper, we present two new approaches for solving slit plane models. One of them simplifies and extends the functional equation approach of the original paper. The other one is inspired by an argument of Lawler; it is more combinatorial, and explains the algebraicity of the product of three series related to the model. It can also be seen as an extension of the classical cycle lemma. Both methods work for any set of steps S. We exhibit a large family of sets S for which the generating function of walks on the slit plane is algebraic, and another family for which it is neither algebraic, nor even D-finite. These examples give a hint at where the border between algebraicity and transcendence lies, and calls for a complete classification of the sets S.Comment: 31 page

Point-plane incidences and some applications in positive characteristic

The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear points, with the extra condition $n< p^2$ if $F$ has characteristic $p>0$. This theorem also underlies a state-of-the-art Szemer\'edi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in $F^d$, for $d=2,3,4$. These are the problem of the minimum number of distinct nonzero values of a non-degenerate bilinear form on a point set in $d=2$, the analogue of the Erd\H os distinct distance problem in $d=2,3$ and additive energy estimates for sets, supported on a paraboloid and sphere in $d=3,4$. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.Comment: A survey, with some new results, for the forthcoming Workshop on Pseudorandomness and Finite Fields in at RICAM in Linz 15-19 October, 2018; 24p

Curiosities and counterexamples in smooth convex optimization

Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy's gradient curves, convergence of Newton's flow,finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka- Lojasiewicz inequality. All examples are planar. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved Ck convex compact sets in the plane, we provide a level set interpolation of a Ck smooth convex function where k 2 is arbitrary. If the intersection is reduced to one point our interpolant has positive denite Hessian, otherwise it is positive denite out of the solution set. Further- more, given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals

Semifields, relative difference sets, and bent functions

Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems