Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

Arrangements in unitary and orthogonal geometry over finite fields

Let V be an n-dimensional vector space over F_q. Let Φ be a Hermitian form with respect to an automorphism σ with σ² = 1. If σ = 1 assume that q is odd. Let $\text{A}$ be the arrangement of hyperplanes of V which are non-isotropic with respect to Φ, and let L be the intersection lattice of $\text{A}$. We prove that the characteristic polynomial of L has n ? v roots 1, q,…, q^{n ? v? 1} where v is the Witt index of Φ.     

Probabilities of incidence between lines and a plane curve over finite fields

We study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree.     

Fluctuations in the number of points on smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8].     

Cyclotomic integers and finite geometry

We obtain an upper bound for the absolute value of cyclotomic integers which has strong implications on several combinatorial structures including (relative) difference sets, quasiregular projective planes, planar functions, and group invariant weighing matrices. Our results are of broader applicability than all previously known nonexistence theorems for these combinatorial objects. We will show that the exponent of an abelian group containing a -difference set cannot exceed where is the number of odd prime divisors of and is a number-theoretic parameter whose order of magnitude usually is the squarefree part of . One of the consequences is that for any finite set of primes there is a constant such that for any abelian group containing a Hadamard difference set whose order is a product of powers of primes in . Furthermore, we are able to verify Ryser's conjecture for most parameter series of known difference sets. This includes a striking progress towards the circulant Hadamard matrix conjecture. A computer search shows that there is no Barker sequence of length with . Finally, we obtain new necessary conditions for the existence of quasiregular projective planes and group invariant weighing matrices including asymptotic exponent bounds for cases which previously had been completely intractable.

     

On a conjecture of Fernando,Hou and Lappano concerning permutation polynomials over finite fields

Let ${\mathbb{F}}_q$ be the finite field with q elements and let $p=\text{char}{\mathbb{F}}_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots +X^{q^a-2}$ is a permutation polynomial of ${\mathbb{F}}_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.     

On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form$a_1{x}_1^{m_1}+a_2{x}_2^{m_2}+\cdots +a_n{x}_n^{m_n}=c.$ We also show that if the value distribution of character sums ${\sum }_{x\in {\mathbb{F}}_q}\chi (a{x}^m+bx)$, $a,b\in {\mathbb{F}}_q$, is known, then one can obtain the number of solutions of the system of equations$\{\begin{array}{cc}\hfill x_1+x_2+\cdots +x_n\hfill & \hfill =\alpha \hfill \\ \hfill x_1^m+x_2^m+\cdots +x_n^m\hfill & \hfill =\beta ,\hfill \end{array}$ for some particular m. We finally apply our results to induce some facts about Waring's problems and the covering radius of certain cyclic codes.     

Counting polynomials over finite fields with prescribed leading coefficients and linear factors

We count the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This is equivalent to counting codewords in Reed-Solomon codes which are at a certain distance from a received word. We first apply the generating function approach, which is recently developed by the author and collaborators, to derive expressions for the number of monic polynomials with prescribed leading coefficients and linear factors. We then apply Li and Wan's sieve formula to simplify the expressions in some special cases. Our results extend and improve some recent results by Li and Wan, and Zhou, Wang and Wang.     

Explicit factors of generalized cyclotomic polynomials and generalized Dickson polynomials of order 2m3 over finite fields

In this paper, we derive explicit factorizations of generalized cyclotomic polynomials and generalized Dickson polynomials of the first kind of order $2^{m}3$, over finite field $F_q$.     

Ambiguity,deficiency and differential spectrum of normalized permutation polynomials over finite fields

We obtain exact formulas for the differential spectrum, deficiency and ambiguity of all normalized permutation polynomials of degree up to six over finite fields.     

An analogue of a theorem of Szüsz for formal Laurent series over finite fields

About 40 years ago, Szüsz proved an extension of the well-known Gauss-Kuzmin theorem. This result played a crucial role in several subsequent papers (for instance, papers due to Szüsz, Philipp, and the author). In this note, we provide an analogue in the field of formal Laurent series and outline applications to the metric theory of continued fractions and to the metric theory of diophantine approximation.     

The theory of non-Abelian tensor fields: Gauge transformations and curvature

We take one more step in formulating the theory of non-Abelian two-tensor fields: we find gauge transformation rules and the curvature tensor for them. To define the theory, we use the surface exponential. We derive a differential equation for the exponential and attempt to formulate its definition as a matrix model. We discuss applications of our construction to the Yang-Baxter equation for integrable models and to string field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 73–91, April, 2006.     

Matching theory and Barnette's conjecture

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian.A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity.As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. (1985) [14]. These allow us to check whether a graph we generated is a brace.