Remarks on the Number of Rational Points on a Class of Hypersurfaces over Finite Fields

Let 𝔽q be the finite field of q elements and f be a nonzero polynomial over 𝔽q. For each b ϵ 𝔽q, let Nq(f = b) denote the number of 𝔽q-rational points on the affine hypersurface f = b. We obtain the formula of Nq(f = b) for a class of hypersurfaces over 𝔽q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.

A classification of isotropic affine hyperspheres

We study affine hypersurfaces [Formula: see text], which have isotropic difference tensor. Note that, any surface always has isotropic difference tensor. In case that the metric is positive definite, such hypersurfaces have been previously studied in [O. Birembaux and M. Djoric, Isotropic affine spheres, Acta Math. Sinica 28(10) 1955–1972.] and [O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417(2) (2014) 918–962.] We first show that the dimension of an isotropic affine hypersurface is either [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Next, we assume that [Formula: see text] is an affine hypersphere and we obtain for each of the possible dimensions a complete classification.

Almost vanishing polynomials and an application to the Hough transform

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.

A SPECIAL DEGREE REDUCTION OF POLYNOMIALS OVER FINITE FIELDS WITH APPLICATIONS

Let f be a polynomial in n variables over the finite field 𝔽q and Nq(f) denote the number of 𝔽q-rational points on the affine hypersurface f = 0 in 𝔸n(𝔽q). A φ-reduction of f is defined to be a transformation σ : 𝔽q[x1, …, xn] → 𝔽q[x1, …, xn] such that Nq(f) = Nq(σ(f)) and deg f ≥ deg σ(f). In this paper, we investigate φ-reduction by using the degree matrix which is formed by the exponents of the variables of f. With φ-reduction, we may improve various estimates on Nq(f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for Nq(f).

RIGIDITY OF AFFINE HYPERSURFACES WITH RANK 1 SHAPE OPERATOR

On a non-degenerate hypersurface it is well known how to induce an affine connection ∇ and a symmetric bilinear form, called the affine metric. Conversely, given a manifold M and an affine connection ∇ one can ask whether this connection is locally realizable as the induced affine connection on a nondegenerate affine hypersurface and to what extend this immersion is unique. In case that the image of the curvature tensor R of ∇ is 2-dimensional and M is at least 3-dimensional a rigidity theorem was obtained in [4]. In this paper, we discuss positive definite n-dimensional affine hypersurfaces with rank 1 shape operator (which is equivalent with 1-dimensional image of the curvature tensor) which are non-rigid. We show how to construct such affine hypersurfaces using solutions of (n - 1)-dimensional differential equations of Monge–Ampère type.