Class numbers of cyclotomic fields

In the first part of the paper we show how to construct real cyclotomic fields with large class numbers. If the GRH holds then the class number h_p⁺ of the pth real cyclotomic field satisfies h_p⁺ > p for the prime p = 11290018777. If we allow n to be composite we have, unconditionally, that $\text{h}{}_{\mathrm{n}}{}^{\mathrm{+}}\text{> n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; \epsilon </mtext>}}$ for infinitely many n. In the second part of the paper we show that if l ?= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then $\text{l}\text{2}$ (l, if l is odd) divides the class number h_n⁺ of the nth cyclotomic field. If the primes are congruent to 1 mod 4l then 2l divides h_n⁺.     

Arithmétique d'une famille de corps cubiques

In this Note, we study the family of polynomials: P(X)=X³?nX²?n, with n=3^sp₁…p_t, where s=0 or 1 and where the p_i, for 1?i?t, are distinct prime numbers and all different from 3, and (4n²+27)/9^s is squarefree. For this family, we determine the arithmetic invariants of the number field $\text{K=}\text{Q}\text{(α),}$ where α is the only real root of the polynomial P(X), and we find the following results: $\text{O}{}_{\mathrm{K}}\text{=}\text{Z}\text{[α]}$ is the ring of integers of K, d_K=?n²(4n²+27) is the discriminant of K; ε=α²+1 is the fundamental unit of O_K and R_K=Log(α²+1) is the regulator of K. To cite this article: O. Lahlou, M. El Hassani Charkani, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Fields with large Kronecker constants

For any algebraic number field K there is a positive number ? such that if α is a nonzero integer of K other than a root of unity, then at least one conjugate of α has absolute value ≥ 1 + ?. It has been conjectured that ? can be taken as $\text{2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1}$, where n is the degree of K over the field of rationals. In this paper various conditions are discussed under which the validity of this conjecture can be established.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

The interval number of a complete multipartite graph

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t closed intervals of the real line $\text{R}$. Trotter and Harary showed that the interval number of the complete bipartite graph K(m,n) is $\text{?}\text{(mn + 1)}\text{(m + n)}\text{?}$. Matthews showed that the interval number of the complete multipartite graph K(n₁,n₂,…,n_p) was the same as the interval number of K(n₁,n₂) when n₁ = n₂ = ? = n_p. Trotter and Hopkins showed that i(K(n₁,n₂,…,n_p)) ≤ 1 + i(K(n₁,n₂)) whenever p ≥ 2 and n₁≥n₂≥ ? ≥n_p. West showed that for each n ≥ 3, there exists a constant c_n so that if p ≥ c_n,n₁ = n²?n ?1, and n₂ = n₃ = ? n_p = n, then i(K(n₁,n₂,…,n_p) = 1 + i(K(n₁, n₂)). In view of these results, it is natural to consider the problem of determining those pairs (n₁,n₂) with n₁ ≥ n₂ so that i(K(n₂,…,n_p)) = i(K(n₁,n₂)) whenever p ≥ 2 and n₂ ≥ n₃ ≥ ? ≥ n_p. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n² ? n ? 1, n) for n ≥ 3 and (7,5).     

Corresponding residue systems in normal extensions

Let K and K′ be number fields with L = K · K′ and F = KφK′. Suppose that $\text{K}\text{F}$ and $\text{K′}\text{F}$ are normal extensions of degree n. Let $\text{B}$ be a prime ideal in L and suppose that $\text{B}$ is totally ramified in $\text{K}\text{F}$ and in $\text{K′}\text{F}$. Let π be a prime element for $\text{B}$_K = $\text{B}$ φ K, and let f(x) ∈ F[x] be the minimum polynomial for π over F. Suppose that $\text{B}$_K · $\text{D}$_L = ($\text{B}$^≠)^e. Then,

$\text{M(B\# : K, K′) =}\text{min}\text{{m, e(t + 1)}}$

, where $\text{t =}\text{min}\text{{t(}\text{K}\text{F}\text{), t(}\text{K′}\text{F}\text{)}}$ and m is the largest integer such that ($\text{B}$_K′)^nm/e φ f($\text{D}$_K′) ≠ {φ}.If we assume in addition to the above hypotheses that [K : F] = [K′: F] = pⁿ, a prime power, and that $\text{B}$ divides p and is totally ramified in $\text{L}\text{F}$, then

$\text{M(B\# : K, K′) ? p}{}^{\mathrm{n?1}}\text{[(p ? 1)(t + p]}$

, with t = t($\text{B}$ : L/F).     

The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

Finite generation of lifted p-adic homology with compact supports. Generalization of the weil conjectures to singular,non-complete algebraic varieties

In Section 1, if $\text{O}$ is a c.d.v.r. with quotient field of characteristic zero and residue class field k, if $\text{A}$ is an $\text{O}$-algebra and if $\text{A =}\text{A}$ ?_$\text{O}$k, then for algebraic families X over A that are polynomially properly embeddable over $\text{A}$, we define the lifted p-adic homology with compact supports$\text{H}{}_{\mathrm{h}}{}^{\mathrm{c}}\text{(X,}\text{A}$2 ?_zQ), which are functors with respect to proper maps. In Section 2, it is shown that, if X is an algebraic variety over k (i.e., if A = k), then the lifted p-adic homology of X with compact supports with coefficients in K is finite dimensional over K = quotient field of $\text{O}$. In Section 3, the results of Sections 1 and 2 are used to generalize both the statement and proof of the Weil “Lefschetz Theorem” Conjecture and the statement (but not the proof) of the Weil “Riemann Hypothesis” Conjecture, to non-complete, singular varieties over finite fields. In addition, the Weil zeta function of varieties over finite fields, is generalized by a device which we call the zeta matrices, W^h(X), 0 ≤ h ≤ 2 dim X, of an algebraic variety X, to varieties over even infinite fields of non-zero characteristic. These are used to give formulas for the zeta functions of each variety in an algebraic family, by means of the zeta matrices of an alebraic family. Sketches only are given. In Section 4, some of the material is duplicated, to define a q-adic homology with compact supports, q ≠ characteristic. The definition only makes sense for algebraic varieties; finite generation is proved. And the Weil “Lefschetz Theorem” Conjecture is established, even for singular, non-complete varieties, as well as a generalization of the Weil “Riemann Hypothesis” Conjecture. (However, zeta matrices do not make sense q-adically.)In Section 5, some special results are proved about p-adic homology with compact supports on affines. And the Weil “Riemann Hypothesis” conjecture is proved p-adically, p = characteristic, for projective, non-singular liftable varieties.     

Graphical t-wise balanced designs

A pair (X, $\text{B}$) will be a t-wise balanced design (tBD) of type t?(v, K, λ) if $\text{B}\text{= (B}{}_{\mathrm{i}}\text{: i ? I)}$ is a family of subsets of X, called blocks, such that: (i) $\text{|X| = v ?}\text{N}$, where $\text{N}$ is the set of positive integers; (ii) $\text{1?t?|B}{}_{\mathrm{i}}\text{|?K?}\text{N}$, for every i ? I; and (iii) if T ? X, |T| = t, then there are λ ? $\text{N}$ indices i ? I where T ? B_i. Throughout this paper we make three restrictions on our tBD's: (1) there are no repeated blocks, i.e. $\text{B}$ will be a set of subsets of X; (2) t ? K or there are no blocks of size t; and (3) $\text{P}{}_{\mathrm{k}}\text{(X)?}\text{B}$ or $\text{B}$ does not contain all k-subsets of X for any t<k?v. Note then that X ? $\text{B}$. Also, if we give the parameters of a specific tBD, then we will choose a minimal K.We focus on the $\text{t?((}\text{p}\text{2}\text{), K, λ)}$ designs with the symmetric group S_p as automorphism group, i.e. X will be the set of $\text{v = (}\text{p}\text{2}\text{)}$ labelled edges of the undirected complete graph K_p and if B ? $\text{B}$ then all subgraphs of K_p isomorphic to B are also in $\text{B}$. Call such tBD's ‘graphical tBD's’. We determine all graphical tBD's with λ = 1 or 2 which will include one with parameters 4?(15,{5,7},1).     

A characterization of the class of credibility matrices corresponding to a certain class of discrete distributions

Recently (see De Vylder & Goovaerts (1984), this issue) so called credibility matrices have been introduced and studied in the framework of general properties of matrices, such as non-negativity, total positivity etc. In the present note we characterize a class of credibility matrices generated by the normed sequence of functions (p_l, p_l,…, p_n) on K = [0, b] where $\text{p}{}_{\mathrm{i}}\text{(θ) =?(i)g(θ)h}{}^{\mathrm{i}}\text{(θ)}$, i=0, …, n, θ ? K, and where ?, g, h are nonnegative (eventually depending on n, n may be finite or infinite). For simplicity we suppose h to be monotonic and continuous.     

Corresponding residue systems in nonnormal extensions of prime degree

Let K₁ and K₂ be number fields and $\text{F = K}{}_1\text{? K}{}_2$. Suppose $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$ are of prime degree p but are not necessarily normal. Let N₁ and N₂ be the normal closures of K₁ and K₂ over F, respectively, L = K₁K₂, N = N₁N₂, and $\text{B}$ be a prime divisor of N which divides p and is totally ramified in $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$. Let ${}_{\mathrm{<mtext>N</mtext><mtext>L</mtext>}}$ be the ramification index of $\text{B}$ in $\text{N}\text{L}$, $\text{t}{}_{\mathrm{<mtext>L</mtext><mtext>F</mtext>}}$ be the total ramification number of $\text{B}$ in $\text{L}\text{F}$, and . Then $\text{M}$(K₁, K₂) is exactly divisible by $\text{B}$^M, where .     

On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

An existence theorem for antichains

Let k₁, k₂,…, k_n be given integers, 1 ? k₁ ? k₂ ? … ? k_n, and let S be the set of vectors x = (x₁,…, x_n) with integral coefficients satisfying 0 ? x_i ? k_i, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities x_i ? y_i, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k₁ + k₂ + … + k_n and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h₁, h₂,…, h_n) which satisfy h₁ + h₂ + … + h_n = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if $\text{l <}\text{K}\text{2}$, $\text{|(}\text{K}\text{2}\text{)H′| = |(}\text{K}\text{2}\text{)H|}$ if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if $\text{l>}\text{K}\text{2}$.     

On pancyclic digraphs

We show that a strongly connected digraph with n vertices and minimum degree ? n is pancyclic unless it is one of the graphs K_p,p. This generalizes a result of A. Ghouila-Houri. We disprove a conjecture of J. A. Bondy by showing that there exist hamiltonian digraphs with n vertices and $\text{1}\text{2}\text{n(n + 1) – 3}$ edges which are not pancyclic. We show that any hamiltonian digraph with n vertices and at least $\text{1}\text{2}\text{n(n + 1) – 1}$ edges is pancyclic and we give some generalizations of this result. As applications of these results we determine the minimal number of edges required in a digraph to guarantee the existence of a cycle of length k, k ? 2, and we consider the corresponding problem where the digraphs under consideration are assumed to be strongly connected.     

On a problem of C. C. Chen and D. E. Daykin

Let α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that $\text{α(k, p, h) ≤ h + 1 + (k ? 1){(p ? h ? 1) × (h}{}^{\mathrm{p}}\text{+ 1)}}{}^{\mathrm{<mtext>1</mtext><mtext>h</mtext>}}$ and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if

$\text{(}\text{n}\text{3}\text{)>(}\text{p}\text{3}\text{)}\text{Σ}\text{i=1}\text{t}\text{(}\text{e}{}_{\mathrm{i}}\text{2}\text{)}$

where e_i is the number of edges of color i, 1 ≤ i ≤ t.     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

Partitions of large multipartites with congruence conditions. II

For 1 ≦ l ≦ j, let $\text{a}$_l = ?_h=1^q(l){a_lh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the a_lh are positive integers such that j > 1, a_l1 < … < a_lq(2) ≦ M, and let $\text{a}$_l^′ = $\text{a}$_l ∪ {0}. Let p(n : $\text{B}$) be the number of partitions of n = (n₁,…,n_j) where, for 1 ≦ l ≦ j, the lth component of each part belongs to $\text{B}$_l and let p¹(n : $\text{B}$) be the number of partitions of n into different parts where again the lth component of each part belongs to $\text{B}$_l. Asymptotic formulas are obtained for p(n : $\text{a}$), p¹(n : $\text{a}$) where all but one n_l tend to infinity much more rapidly than that n_l, and asymptotic formulas are also obtained for p(n : $\text{a}$′), p¹(n ; $\text{a}$′), where one n_l tends to infinity much more rapidly than every other n_l. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the n_l tend to infinity at approximately the same rate.