A complete hypergeometric point count formula for Dwork hypersurfaces

We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ over a finite field $$\mathbb {F}_p$$ . Salerno expresses the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of quotients of p-adic gamma functions under the condition that $$d|p-1$$ . In this paper, we first express the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of McCarthy’s p-adic hypergeometric series for any odd prime p not dividing $$d(d-1)$$ , and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene’s finite field hypergeometric series.

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

In this paper, we calculate the Weiner index, some degree-based topological indices, and some eccentricity-based topological indices of Join graph of Γ(F1 ×F2 ×F3) and Γ(J1 ×J2 ×J3), where F1, F2, F3, J1, J2, J3 are finite fields of order at least two. The vertex set of Γ(F1×F2×F3) (Γ(J1×J2 ×J3)) is Z∗(F1 ×F2 ×F3) (Z∗(J1 ×J2 ×J3)), the set of non-zero zero divisors, and two distinct vertices (x1, x2, x3), (y1, y2, y3) are adjacent if and only if (x1, x2, x3) · (y1, y2, y3) = (0, 0, 0), the additive identity of the ring F1 × F2 × F3 (J1 × J2 × J3). The vertex set of Join graph Γ(F1 × F2 × F3) + Γ(J1 × J2 × J3) is V (Γ(F1 × F2 × F3)) ∪ V (Γ(J1 × J2 × J3)) and edge set is E(Γ(F1 × F2 × F3)) ∪ E(Γ(J1 × J2 × J3)) ∪ {(x1, x2, x3) · (y1, y2, y3) : (x1, x2, x3) ∈ V (Γ(F1 × F2 × F3)), (y1, y2, y3) ∈ V (Γ(J1 × J2 × J3))}.

Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2∈F‾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(L1,L2)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(L1,L2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1].

Let be a finite field of order q, t be a prime and be positive integers. In this article, we find all irreducible divisors of over where and such that are distinct odd primes and c is a positive integer with and Moreover, we construct an a-constacyclic code by using these irreducible divisors.

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

Commutative presemifields and semifields

One of the most celebrated applications of Gauss’s 2 F 1 _2F_1 hypergeometric functions is in connection with the rapid convergence of sequences and special values that arise in the theory of arithmetic and geometric means. This theory was the inspiration for a recent paper (see M. J. Griffin, Ken Ono, N. Saikia, and W.-L. Tsai, [AGM and jellyfish swarms of elliptic curves, Amer. Math. Monthly 130 (2023), no. 4, 355-369]) in which a finite field analogue of A G M R \mathrm {AGM}_\mathbb {R} was defined and then studied using finite field hypergeometric functions. Instead of convergent sequences, one gets directed graphs that combine to form disjoint unions of graphs that individually resemble jellyfish. Echoing the connection of hypergeometric functions to periods of elliptic curves, these graphs organize elliptic curves over finite fields. Here we use such “jellyfish swarms” to prove new identities for Gauss’s class numbers of positive definite binary quadratic forms. Moreover, we prove that the sizes of jellyfish are in part dictated by the order of the prime above 2 in certain class groups.

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

For non-negative integers \(l_{1}, l_{2},\ldots , l_{n}\), we define character sums \(\varphi _{(l_{1}, l_{2},\ldots , l_{n})}\) and \(\psi _{(l_{1}, l_{2},\ldots , l_{n})}\) over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene’s finite field hypergeometric series. We then express the number of points on the hyperelliptic curves \(y^2=(x^m+a)(x^m+b)(x^m+c)\) and \(y^2=x(x^m+a)(x^m+b)(x^m+c)\) over a finite field in terms of the character sums \(\varphi _{(l_{1}, l_{2}, l_{3})}\) and \(\psi _{(l_{1}, l_{2}, l_{3})}\), and finally obtain expressions in terms of the finite field hypergeometric series.

Motivated by applications to cryptography, mathematicians have employed p p -adic cohomological methods to compute the zeta functions of various classes of varieties defined over finite fields of order q = p a q=p^a in an amount of time polynomial in a a , assuming the characteristic p p is fixed. The varieties that are considered generally have smooth representations in either affine or projective space. Extending the methods of Gaudry and Gürel, we present a polynomial-time algorithm which computes the zeta function of singular superelliptic curves, and provide the results of an implementation in MAGMA. Assuming the curve has geometric genus g g , and that the characteristic p p is fixed, the running time of the algorithm is O ( a 3 + ε g 5 + ε ) O(a^{3+\varepsilon }g^{5+\varepsilon }) .

In 2010, Shpilka and Volkovich established a prominent result on the equivalence of polynomial factorization and identity testing. It follows from their result that a multilinear polynomial over the finite field of order 2 can be factored in time cubic in the size of the polynomial given as a string. Later, we have rediscovered this result and provided a simple factorization algorithm based on computations over derivatives of multilinear polynomials. The algorithm has been applied to solve problems of compact representation of various combinatorial structures, including Boolean functions and relational data tables. In this paper, we describe an improvement of this factorization algorithm and report on preliminary experimental analysis.

The minimum volume of subspace trades

Classification of some permutation quadrinomials from self reciprocal polynomials over [formula omitted