The class number of cyclotomic function fields

Let k be a rational function field over a finite field. Carlitz and Hayes have described a family of extensions of k which are analogous to the collection of cyclotomic extensions {Q(ζm)| m ≥ 2} of the rational field Q. We investigate arithmetic properties of these “cyclotomic function fields.” We introduce the notion of the maximal real subfield of the cyclotomic function field and develop class number formulas for both the cyclotomic function field and its maximal real subfield. Our principal result is the analogue of a classical theorem of Kummer which for a prime p and positive integer n relates the class number of Q(ζpn + ζpn?1), the maximal real subfield of Q(ζpn), to the index of the group of cyclotomic units in the full unit group of Z[ζpn].     

On the prime power factorization of n!

For a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For two fixed integers m?2 and r with 0?r?m−1, let A(x;m,q,r) denote the numbers of positive integers n?x for which . In this paper we shall prove a sharp asymptotic formula of A(x;m,q,r).     

Power integral bases in prime-power cyclotomic fields

Let p be an odd prime and q=pm, where m is a positive integer. Let ζ be a primitive qth root of unity, and Oq be the ring of integers in the cyclotomic field Q(ζ). We prove that if Oq=Z[α] and , where is the class number of Q(ζ+ζ⁻¹), then an integer translate of α lies on the unit circle or the line Re(z)=1/2 in the complex plane. Both are possible since Oq=Z[α] if α=ζ or α=1/(1+ζ). We conjecture that, up to integer translation, these two elements and their Galois conjugates are the only generators for Oq, and prove that this is indeed the case when q=25.     

Congruences Involving Bernoulli Numbers and Fermat-Euler Quotients

Let B_m be the mth Bernoulli number in the even suffix notation and let q(a, n)=(a^?(n)−1)/n be the Fermat-Euler quotient, where a, n?2 are relatively prime positive integers and ? is the Euler totient function. The main purpose of this paper is to devise a certain congruence involving the Bernoulli number and Fermat-Euler quotient, which leads to several important arithmetic properties of Bernoulli numbers.     

Torsion points on the jacobian of a Fermat quotient

Let p≥5 be a prime, ζ a primitive pth root of unity and λ=1−ζ. For 1≤s≤p−2, the smooth projective model C_p,s of the affine curve v^p=u^s(1−u) is a curve of genus (p−1)/2 whose jacobian J_p,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ). In 1981, Greenberg determined the field of rationality of the p-torsion subgroup of J_p,s and moreover he proved that the λ³-torsion points of J_p,s are all rational over Q(ζ). In this paper we determine quite explicitly the λ³-torsion points of J_p,1 for p=5 and p=7, as well as some further p-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for pth powers.     

A specific family of cyclotomic polynomials of order three

Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p q r and 2r ≡ 1(mod pq),we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1(mod 3).     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold: (1) For any rational prime q and any positive rational integer m. algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q^m. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set \(\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P \ne q\) is any prime such that q^{m +1} ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions.We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable.     

Représentation de −1 comme somme de carrés dans un corps cyclotomique quelconque

Let m ≥ 3 be an odd integer, and let K^(m) = Q(e^2πi/m) be the cyclotomic field of the m-th roots of unity. Then s(K^(m)) (the “stufe” of K^(m), that is to say, the smallest number of squares necessary to represent ?1 in K^(m) is equal to 2 or to 4 depending on whether the multiplicative order of 2 modulo m is even or odd.     

Waring's number mod m

Let p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer is a sum of s kth powers . We establish γ(k,pn)?[k/2]+2 and provided that k is not divisible by (p−1)/2. Next, let t=(p−1)/(p−1,k), and q be any positive integer. We show that if ?(t)?q then γ(k,pn)?c(q)k1/q for some constant c(q). These results generalize results known for the case of prime moduli.

Video abstract

For a video summary of this paper, please visit http://www.youtube.com/watch?v=zpHYhwL1kD0.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

On the coefficients of ternary cyclotomic polynomials

We study coefficients of ternary cyclotomic polynomials Φ_pqr(z)=∏_ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.     

Stably extendible tangent bundles over lens spaces

The purpose of this paper is to study the stable extendibility of the tangent bundle τn(p) over the (2n+1)-dimensional standard lens space Ln(p) for odd prime p. We investigate for which m the tangent bundle τn(p) is stably extendible to Lm(p) but is not stably extendible to Lm+1(p), where we consider m=∞ if τn(p) is stably extendible to Lk(p) for any k?n, and determine m in the case n?p−3.     

Inverse cyclotomic polynomials

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x)=(xn−1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Ψn(x) are ?1 in absolute value. We establish various properties of the coefficients of Ψn(x), especially focusing on the easiest non-trivial case where n is composed of 3 distinct odd primes.     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp₁(±x)Φp₂(±xp₁)?Φpr(±xp₁p₂?pr−1), where N=p₁p₂?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α²pβ−1 with odd prime p or separable polynomials of any odd degree.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

On the order of an automorphism of a smooth hypersurface

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d ? 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d ? 1) ⁿ⁺¹; and if p > (d ? 1) ⁿ then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φ _n+2(1 ? d) where Φ _n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.     

On the prime power factorization of n!, II

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems.     

Uniform cyclotomy

Let p be the characteristic of the finite field GF(q), and let e be a divisor of q?1, e≥3. We determine the cyclotomic numbers of order e over GF(q) for the case where ?1 is a power of p modulo e. In this case most of the cyclotomic numbers are equal. We also prove a theorem about difference sets.