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.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Some contributions to the theory of cyclic quartic extensions of the rationals

K is a cyclic quartic extension of Q iff $\text{K = Q((rd + p d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>)</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where d > 1, p and r are rational integers, d squarefree, for which p² + q² = r²d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, $\text{d(}\text{K}\text{Q}\text{)}$, of the general cyclic quartic extension is given by $\text{d(}\text{K}\text{Q}\text{) = (W}{}^2\text{d}{}^2\text{)}$ for an explicitly computable rational integer W. We next find that the relative discriminant, $\text{d(}\text{K}\text{F}\text{)}$, is given by $\text{d(}\text{K}\text{F}\text{) = (W d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where $\text{F = Q (d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is K′s uniquely determined quadratic subfield. We use this last result in conjunction with Corollary 3, page 359, of Narkiewicz's “Elementary and Analytic Theory of Algebraic Numbers” (PWN-Polish Scientific Publishers, 1974) to establish the following Theorem 1: If the (wide) class number of$\text{F = Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$is odd then every cyclic quartic extensionKofQcontainingFhas a relative integral basis overF. We give a second, more organic, proof of Theorem 1 which also allows us to prove the following converse result, namely Theorem 2: Suppose the quadratic fieldFis contained in some cyclic quartic extension ofQand suppose thatFhas even (wide) class number. There then is a cyclic quartic extensionKofQcontainingFsuch thatKhas no relative integral basis overF.     

On subgraph number independence in trees

For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If $\text{F}$ and $\text{G}$ are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F∈$\text{F}$, are linearly independent when G is restricted to $\text{G}$. For example, if $\text{F}$ = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and $\text{F}$ is the family of all (finite) trees, then of course N_K1(T) ? N_K2(T) = 1 for all T∈$\text{F}$. Slightly less trivially, if $\text{F}$ = {S_n: n = 1, 2, 3,…} (where S_n denotes the star on n edges) and $\text{G}$ again is the family of all trees, then Σ_n=1^∞(?1)ⁿ⁺¹N_Sn(T)=1 for all T∈$\text{G}$. It is proved that such a linear dependence can never occur if $\text{F}$ is finite, no F∈$\text{F}$ has an isolated point, and $\text{G}$ contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).     

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).     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

When is a Pixley-Roy hyperspace CCC?

A space X is said to satisfy condition (C) if for every Y?X with |Y|=ω₁, any family $\text{G}$ of open subsets of Y with |$\text{G}$|=ω₁ has a countable network. It is easy to see that if X satisfies condition (C), then its Pixley-Roy hyperspace $\text{F}$[X] is CCC. We show that under MA_ω1 condition (C) is also necessary for $\text{F}$[X] to be CCC, but under CH it is not.     

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 .     

Asymptotics of functions satisfying the γ-weak inequality with applications to uniformly bounded representations

Let G be a real reductive group of class $\text{H}$, and π a uniformly bounded representation of G on a Hilbert space having infinitesimal character. We then show that the K-finite matrix elements of π decay “exponentially” on G provided that the infinitesimal character of π is in general position. Further we show that π is infinitesimally equivalent to a subquotient of a cuspidal principal series representation π_Q,ω,ν where ν belongs to a tube domain defined by ?Q. These facts follow from the asymptotics of functions satisfying the γ-weak inequality.     

On the representation theory for full decomposable forms

Let $\text{k}\text{Q}$ be any finite normal extension and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. The ring class field K corresponding to D is normal over Q. We solve the problem of determining which full decomposable forms associated to the invertible ideals of D integrally represent a given positive integer m. First we establish a one-to-one correspondence between the improperly equivalent classes of such forms and the conjugacy classes of $\text{G(}\text{K}\text{Q}\text{)}$ which are contained in $\text{G(}\text{K}\text{k}\text{)}$. The solution is then seen to rest upon determining the Artin class in $\text{G(}\text{K}\text{Q}\text{)}$ of the unramified primes dividing m. This is accomplished by evaluating certain induced characters of $\text{G(}\text{K}\text{Q}\text{)}$ congruentially in terms of an associated integer linear recurrence sequence.     

Automorphisms and holomorphic differentials in characteristic p

Let G be a group of automorphisms of a function field F of genus g_F over an algebraically closed field K. The space $\text{Ω}{}_{\mathrm{F}}$ of holomorphic differentials of F is a g_F^? dimensional K-space. In response to a query of Hecke, Chevalley and Weil (Abh. Math. Sem. Univ. Hamburg, 10 (1934), 358–361) completely determined the structure of $\text{Ω}{}_{\mathrm{F}}$ as representation space for G in the classical case. They carried out the proof for the special case in which F is unramified over the fixed field of G. The case of cyclic ramified extensions had been previously considered by Hurwitz (Math. Ann., 41 (1893), 37–45). Weil (Abh. Math. Sem. Univ. Hamburg, 11 (1935), 110–115) gave a proof in the general case. The treatment in the last two papers is analytical. In characteristic p, the problem is open. If G is cyclic and F is unramified over the fixed field E of G, Tamagawa (Proc. Japan Acad., 27 (1951), 548–551) proved that the representation is the direct sum of one identity representation of degree 1 and g_E ? 1 regular representations. The principal object of this paper is an extension of Tamagawa's result to arbitrary cyclic extensions of p-power degree. The number of times an indecomposable representation of given degree occurs in the representation of G on $\text{Ω}{}_{\mathrm{F}}$ is explicitly determined in terms of g_E and the Witt vector characterizing the extension $\text{F}\text{E}$. The paper also contains a purely algebraic proof of the result of Chevalley and Weil for arbitrary cyclic extensions of degree relatively prime to p. Using character theory, it can be extended to arbitrary groups of order relatively prime to the characteristic.     

The additive and logical complexities of linear and bilinear arithmetic algorithms

Let C_A(±) be the additive complexity of a (bi)linear algorithm A for a given problem; D(A) and $\text{D}\text{(A)}$ are two acyclic diagraphs that represent A, each of them is obtained from another one by reversing directions of all edges; ir(D) and do(D) are two numbers that are introduced to measure the structural deficiencies of an acyclic digraph D. K and Q are the numbers of outputs and input-variables. $\text{do(}\text{D}\text{(A))}$, do(D(A)), and ir(D(A)) characterize the logical complexity of A. It is shown that C_A(±) + do(D(A)) + ir(D(A)) = ω(K + Q)log(K + Q) and $\text{C}{}_{\mathrm{A}}\text{(±) + do(}\text{D}\text{(A)) = ω(K + Q)}\text{log}\text{(K + Q)}$ in the cases of DFT, vector convolution, and matrix multiplication. Also lower bounds on C_A(±) + do(D(A)) and on C_A(±) are expressed in terms of algebraic quantities such as the ranks of matrices and of multidimensional tensors associated with the problems.     

Synergy of homomorphisms in relational systems

Suppose that a relational system $\text{R}$ can be exhausted (in an obvious sense) by a family R_ω, $\text{ω ? Ω}$, of subrelational systems, each of which can be mapped by a homomorphism onto a subrelational system S_ω of a second relational system $\text{S}$. We show that, under suitable finiteness conditions, there is a homomorphism from $\text{R}$ into $\text{S}$ which finitely agrees with the homomorphisms mapping R_ω onto S_ω. A similar result holds for isomorphisms.     

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.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

On the analytic continuation of a certain Dirichlet series

A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by

$\text{D}{}_{\mathrm{F}}\text{(s,p,α)}\text{=}\text{∑}\text{α∈Z}{}^{\mathrm{n}}\text{?{0}}\text{F(α)}{}^{\mathrm{?s}}\text{e(ρF(α)+〈α, α〉)}$

where ? ∈ $\text{Q}$, α ∈ $\text{Q}$ⁿ, 〈x, y〉 = x₁y₁ + ? + x_ny_n, e(a) = exp (2πia) for a ∈ $\text{R}$, and s = σ + ti is a complex number. The author proves that: (1) D_F(s, ?, α) has analytic continuation into the whole s-plane, (2) D_F(s, ?, α), ? ≠ 0, is a meromorphic function with at most a simple pole at $\text{s =}\text{n}\text{δ}$. The residue at $\text{s =}\text{n}\text{δ}$ is given explicitly. (3) ? = 0, α ? $\text{Z}$ⁿ, D_F(s, 0, α) is analytic for $\text{α>,}\text{n}\text{(δ ? 1)}$.     

A fixed point theorem for eventually nonexpansive semigroups of mappings

Let K be a subset of a Banach space X. A semigroup $\text{F}$ = {?_α ∥ α ∞ A} of Lipschitz mappings of K into itself is called eventually nonexpansive if the family of corresponding Lipschitz constants $\text{{k}{}_{\mathrm{\alpha }}\text{¦ α ? A}}$ satisfies the following condition: for every ? > 0, there is a γ?A such that k_β < 1 + ? whenever ?_β ? ?_γ$\text{F}$ = {?_gg?_α ¦ ?_α ? $\text{F}$}. It is shown that if K is a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $\text{F}$:K → K is an eventually nonexpansive, commutative, linearly ordered semigroup of mappings, then $\text{F}$ has a common fixed point. This result generalizes a fixed point theorem by Goebel and Kirk.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.