Primitive polynomial with three coefficients prescribed

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a₁,a₂,a₃)F_q³, there exists a primitive polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿσ_n over F_q of degree n with the first three coefficients σ₁,σ₂,σ₃ prescribed as a₁,a₂,a₃ when n8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over F_q with the first three coefficient prescribed where the characteristic of F_q is 2 or 3.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Primitive normal polynomials with the specified last two coefficients

We get that for any element a∈F_q and any primitive element , there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹+?+(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n with σ_n−1=a, σ_n=b for n≥5. The estimates of hybrid extended Kloostermann sums over finite fields and some new variations of Cohen’s sieve techniques help to get the above result.     

特征2的有限域中一些特殊元素的存在性(II)

设q=2s.s,n为正整数,Fqn为qn元素的有限域.在本文中,我们考虑Fqn中一些特殊元素的存在性.主要结果是:当下面的条件之一成立时,在Fqn中存在ξ使得ξ和ξ+ξ-1都是本原元并且ξ+ξ-1还是一个正规元:1.当n|(q-1)时,n37,s>6,或者2.当n|■(q-1)时,n≥34,s>6.进一步,如果n是奇数,则当下列条件之一成立时,存在ξ∈Fqn使得ξ和ξ+ξ-1都是Fqn的本原正规元:1.当n|(q-1)时,n≥257,s>9,或者2.当n■(q-1)时,n≥43,s≥9.     

Primitive complete normal bases: Existence in certain 2-power extensions and lower bounds

The present paper is a continuation of the author’s work (Hachenberger (2001) [3]) on primitivity and complete normality. For certain 2-power extensions E over a Galois field F_q, we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of E/F_q. The main result is as follows: Letq≡3mod4and letm(q)≥3be the largest integer such that2^m(q)dividesq²−1; ifE=F_q^2l, wherel≥m(q)+3, then there exists a primitive element inEthat is completely normal overF_q.Our method not only shows existence but also gives a fairly large lower bound on the number of primitive completely normal elements. In the above case this number is at least 4⋅(q−1)^2l−2. We are further going to discuss lower bounds on the number of such elements in r-power extensions, where r=2 and q≡1mod4, or where r is an odd prime, or where r is equal to the characteristic of the underlying field.     

On the football pool problem for 6 matches: A new upper bound

Consider the set (F₃)⁶ of all 6 tuples x=(x₁,…, x₆) with x_i? {0, 1, 2}. We show that there exists a subset $\text{T}$ of (F₃)⁶ with 79 elements such that for any x of (F₃)⁶ there is an element if $\text{T}$ which differs from x in at most one coordinate.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

The number of polynomial functions which permute the matrices over a finite field

Let F = GF(q) denote the finite field of order q, and let F_n×n denote the algebra of n × n matrices over F. A function f:F_n×n → F_n×n is called a scalar polynomial function if there exists a polynomial f(x) ?F[x] which represents f when considered as a matrix function under substitution. In this paper a formula is obtained for the number of permutations of F_n×n which are scalar polynomial functions.     

More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.     

Conics and caps

In this article, we begin with arcs in PG(2, q ⁿ ) and show that they correspond to caps in PG(2n, q) via the André/Bruck?CBose representation of PG(2, q ⁿ ) in PG(2n, q). In particular, we show that a conic of PG(2, q ⁿ ) that meets ?_?? in x points corresponds to a (q ⁿ ?+?1 ? x)-cap in PG(2n, q). If x?=?0, this cap is the intersection of n quadrics. If x?=?1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics.     

M-P invertible matrices and unitary groups over Fq 2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

ADDITIVE FAMILIES OF INVARIANTS OF FORMS OF DEGREE d

Abstract

Let d be a positive integer, and F be a field of characteristic zero. Suppose that for each positive integer n, I _n, is a GL _n,(F)- invariant of forms of degree d in x₁, …, x _n, over F. We call {I _n} an additive family of invariants if I _p+q (f ⊥ g) = I _p(f).I _q(g) whenever f; g are forms of degree d over F in x _l, …, x _p; …, x _q respectively, and where (f ⊥ g)(x _l, …, x _p+q) = f(x ₁, …, x _p,) + g (x _p+1, …, x _p+q). It is well-known that the family of discriminants of the quadratic forms is additive. We prove that in odd degree d each invariant in an additive family must be a constant. We also give an example in each even degree d of a nontrivial family of invariants of the forms of degree d. The proofs depend on the symbolic method for representing invariants of a form, which we review.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Maximum IPP Codes of Length 3

Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let ${D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let D(a, b) = {(x₁, . . . , x_n) ? Qⁿ : x_i ? {a_i, b_i} for 1 ￡ i ￡ n}{D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}. Let C^* = è_{a, b ? C}D(a, b){C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}. The code C is said to have the identifiable parent property (IPP) if, for any x ? C^*{{\rm {\bf x}} \in C^*}, ?_{x ? D(a, b)}{a, b} 1 ?{{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} . Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.T? and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that 3q + 6 - 6 é?{q+1}ù ￡ F(3, q) ￡ 3q + 6 - é6 ?{q+1}ù{3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}, and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r ² + 2r or r ² + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any x ? C^*{{\rm {\bf x}} \in C^*}, one can find, in constant time, a ? C{{\rm {\bf a}} \in C} such that if x ? D (c, d){{\rm {\bf x}} \in D ({\bf c, d})} then a ? {c, d}{{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}.     

Analytic extension of non quasi-analytic Whitney jets of Roumieu type

Let (M_r)_{r∈? 0} be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function ? on ?ⁿ belonging to the non quasi-analytic (M_r)-class of Roumieu type, there is an element g of the same class which is analytic on ?ⁿ F and such that D^α ?(x) = D^αg(x) for every σ ∈ ?₀ ⁿ SBAP and x ∈ F.     

Connection-graph and iteration-graph of monotone boolean functions

In this paper, we establish some relationships between the circuits of the connection-graph G_C(F), and the circuits of theiteration-graph G₁(F), of a monotone boolean function F. We first show that if G₁(F) contains an element circuit of length multiple of p? {2,3}, then G_C(F) contains an elementary circuit of length multiple of p; then we prove that if G_C(F) is a subgraph of a caterpillar, then G₁(F) is a subgraph of a tree; at last we exhibit an infinite family of monotone boolean functions {F_n; n = 2 × 5^q, q ≥ 1} such that any G_C(F_n) is a subgraph of a tree, and G₁(F_n) contains a circuit of length 2^q+1, i.e., of the order $\text{n}{}^{\mathrm{<mtext>log</mtext><mtext>2</mtext><mtext>log5</mtext>}}$.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let $\text{?(x)?F[x]}$. Then f(x) defines, via substitution, a function from F_n×n, the n×n matrices over F, to itself. Any function $\text{?:F}{}_{\mathrm{n\times n}}\text{→ F}{}_{\mathrm{n\times n}}$ which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on F_n×n. After first determining the number of scalar polynomial functions on F_n×n, the authors find necessary and sufficient conditions on a polynomial $\text{?(x) ? F[x]}$ in order that it defines a permutation of (i) $\text{D}$_n, the diagonalizable matrices in F_n×n, (ii)$\text{R}$_n, the matrices in F_n×n all of whose roots are in F, and (iii) the matric ring F_n×n itself. The results for (i) and (ii) are valid for an arbitrary field F.