Over a fixed finite field ${\bf F}_p$, families ofpolynomial equations$f_i(x_1, \dots, x_{n_N}) = 0$ for $i = 1, \dots, k_N$,that are uniformly determined by a parameter $N$, areconsidered. The notion of a uniform family is defined interms of first-order logic.A notion of an abstract Euler characteristic is used togive sense to a statement that the system has a solutionfor infinite $N$, and a statement linking the solvabilityof a linear system for infinite $N$ with its solvabilityfor finite $N$ is proved.This characterisation is used to formulate a criterionyielding degree lower bounds for various ideal membership proof systems (for example,Nullstellensatz and the polynomial calculus).Further, several results about Euler structures(structures with an abstract Euler characteristic) are proved, and the case of fields, in particular,is investigated more closely. 1991 Mathematics Subject Classification: primary 03F20, 12L12, 15A06;secondary 03C99, 12E12, 68Q15, 13L05.

We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model complete and o-minimal structure which is exponentially bounded, and in which the Gamma function on the positive real line is definable. 2000 Mathematics Subject Classification: primary 03C10, 32B05, 32B20; secondary, 26E05.

This paper contains results on the structure of generic local polar curves $P(\tau)$ of a reduced germ of a complex analytic plane curve $C$. We first prove a decomposition theorem of the branches of $P(\tau)$ into bunches. By construction, all the branches in the same bunch have the same contact with all branches of $C$. It follows that the Puiseux expansions of all the branches of a given bunch coincide with that of a branch of $C$ up to an order depending upon the bunch. An initial part of these expansions is therefore independant of $\tau$. In the second part, we study to what extent the contact of the branches of $C$ with the branches of $P(\tau)$ determines the topological type of $C$. We build from these contacts a matrix which is determined by the topological type of $C$ and determines it. In an appendix, we explain how to recover, from the results of the first part, the L\^e--Michel--Weber theorem on the behaviour of polar curves in an embedded resolution of singularities of $C$. Cet article pr\'esente des r\'esultats sur la structure des courbes polaires locales g\'en\'eriques $P(\tau)$ d'un germe r\'eduit de courbe analytique complexe plane $C$. Nous d\'emontrons d'abord un th\'eor\`eme de d\'ecomposition en paquets des branches de $P(\tau)$. Par construction, toutes les branches d'un m\^eme paquet ont le m\^eme contact avec chacune des branches de $C$. L'ensemble de ces paquets est index\'e par un graphe qui ne d\'epend que de la topologie de la courbe $C$ donn\'ee. En cons\'equence, le d\'eveloppement de Puiseux de toutes les branches d'un m\^eme paquet de $P(\tau)$ coincide avec celui d'une branche de $C$ jusqu'\`a un ordre d\'ependant du paquet. Une partie initiale de ce d\'eveloppement est donc ind\'ependante de $\tau$. Dans la deuxi\`eme partie de ce travail, nous \'etudions dans quelle mesure le contact avec les branches de $C$ des branches de $P(\tau)$ d\'etermine le type topologique de $C$. Nous construisons \`a partir de tous ces contacts une matrice qui ne d\'epend que du type topologique de $C$ et le d\'etermine. Dans un appendice, nous montrons comment retrouver, \`a partir des r\'esultats prouv\'es dans la premi\`ere partie, le th\'eor\`eme de L\^e--Michel--Weber sur le comportement des polaires dans une r\'esolution plong\'ee des singularit\'es de $C$. E-mail: [email protected] 1991 Mathematics Subject Classification: 14H20, 32S10.

The relation between $\mathbb{Q}$-curves and certainabelian varieties of $\operatorname{GL}_2$-type was established by Ribet (`Abelian varieties over $\mathbb{Q}$ and modular forms', {\em Proceedings ofthe KAIST Mathematics Workshop} (1992) 53--79)and generalized to building blocks, the higher-dimensional analogues of $\mathbb{Q}$-curves,by Pyle in her PhD Thesis (University of California at Berkeley, 1995).In this paper we investigate some aspectsof $\mathbb{Q}$-curves with no complex multiplicationand the corresponding abelian varieties of $\operatorname{GL}_2$-type, for which we mainly use the ideas and techniques introduced by Ribet(op. cit. and `Fields of definition of abelian varieties with real multiplication', {\em Contemp.\ Math.} 174(1994) 107--118).After the Introduction,in Sections 2 and 3 we obtain a characterization of thefields where a $\mathbb{Q}$-curve and all the isogeniesbetween its Galois conjugates can be defined up to isogeny, and we apply it to certain fields of type $(2,\dots,2)$.In Section 4 we determine the endomorphism algebras of all the abelian varieties of $\operatorname{GL}_2$-type having as a quotient a given $\mathbb{Q}$-curve in easily computable terms.Section 5 is devoted to a particular case ofWeil's restriction of scalars functor applied to a $\mathbb{Q}$-curve,in which the resulting abelian variety factors over $\mathbb{Q}$ up to isogeny as a product of abelianvarieties of $\operatorname{GL}_2$-type.Finally, Section 6 contains examples:we parametrize the $\mathbb{Q}$-curves coming fromrational points of the modular curves $X^*(N)$ having genus zero, and we apply the results of Sections 2--5 to some of the curves obtained.We also give results concerning the existence of quadratic $\mathbb{Q}$-curves. 1991 Mathematics Subject Classification:primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.

Theorem 1 asserts that \emph{in a finitely generated prosoluble group, every subgroup of finite index is open}. This generalises an old result of Serre about pro-$p$ groups. It follows by a standard argument from Theorem 2: \emph{in a $d$-generator finite soluble group, every element of the derived group is equal to a product of $72d^2 +46d$ commutators}. This result about finite soluble groups is proved by induction on the order of the group, and is elementary though rather intricate. The essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field. Corollaries include the following. \emph{Let $\Gamma$ be a finitely generated prosoluble group. Then each term of the lower central series of $\Gamma$ and each power subgroup $\Gamma ^n$ is closed}. 1991 Mathematics Subject Classification: 20E18, 20D10.

This paper is devoted to the complete classification of integrable one-component evolution equations whose field variable takes its values in an associative algebra. The proof that the list of non-commutative integrable homogeneous evolution equations is complete relies on the symbolic method. Each equation in the list has infinitely many local symmetries and these can be generated by its recursion (recursive) operator or master symmetry. 1991 Mathematics Subject Classification: 13A50, 16-XX, 22E70, 35A30, 35Q53, 58F07.

We introduce a variant of the usual K\"{a}hler forms on singular free divisors, and show that it enjoys the same depth properties as K\"{a}hler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the Gauss--Manin connection, in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. This applies in particular to the family $\{D(f_\lambda)\}_{\lambda\in \Lambda}$ of discriminants of a versal deformation $\{f_\lambda\}_{\lambda\in\Lambda}$ of a singularity of a mapping. 1991 Mathematics Subject Classification: 14B07, 14D05, 32S40.

Let $H=k\cal Q$ be a finite-dimensional connected wild hereditary path algebra, over some field $k$. Denote by $H$-reg the category of finite-dimensional regular $H$-modules, that is, the category ofmodules $M$with $\tau_H^{-m}(\tau_H^m M) \cong M$ for all integers $m$, where $\tau_H$ denotes the Auslander--Reiten translation. Call a filtration\begin{equation}M = M_0\supset M_1\supset\ldots\supset M_r\supset M_{r+1}=0 \tag{$*$}\end{equation}of a regular $H$-module $M$ a {\em regular filtration}if all subquotients $M_i/M_{i+1}$ are regular. Call a regular filtration $(*)$ a {\em regular composition series} if it is strictly decreasing and has no properrefinement.A regular component $\cal C$ in the Auslander--Reiten quiver $\Gamma (H)$ of $H$-modis called {\em filtration closed} if, for each$M\in\text{add\,}\cal C$, the additive closure of $\cal C$, and each regular filtration $(*)$ of $M$, all the subquotients $M_i/M_{i+1}$ are also in$\text{add\,}\cal C$. We show that mostwild hereditary algebras have filtration-closedAuslander--Reiten components. Moreover,we deduce from this that there are also {\em almost serial} components, that is regular components $\cal C$, such that any indecomposable $X\in\cal C$ has a unique regular composition series.This composition series coincides with theAuslander--Reiten filtration of $X$, given by the maximal chain of irreducible monos ending at $X$. 1991 Mathematics Subject Classification: 16G70, 16G20, 16G60, 16E30.

Suppose that $N$ is a normal $p$-subgroup of a finite group $G$ and let $G^0$ be the set of elements of $G$ whose $p$-part lies in $N$. We prove the existence of a canonical basis ${\rm IBr}(G, N)$ of the space of complex class functions of $G$ defined on $G^0$, such that the restriction $\chi^0$ of any irreducible complex character $\chi$ of $G$ is a linear combination $\sum_{\phi\in{\rm IBr}(G, N)} d_{\chi \phi} \phi$ of the elements of this basis, where the $d_{\chi \phi}$ are non-negative integers. Furthermore, if we write $\Phi_\phi=\sum_{\chi} d_{\chi \phi}\chi$, then the $\Phi_\phi$ form the K\"ulshammer--Robinson ${\Bbb Z}$-basis of the ${\Bbb Z}$-module generated by the characters afforded by the $N$-projective $RG$-modules, where $R$ is a certain complete discrete valuation ring. By using these `decomposition numbers', it is possible to define a linking in the set of the irreducible complex characters of $G$. 1991 Mathematics Subject Classification: 20C15, 20C20.

The authors consider the system of forced differential equations with variable delays $$x'(t) + \sum^N_{j=1}B_j(t)x(t-\tau_j(t)) = F(t)\eqno(*)$$ where $B_j(t)$ is a continuous $n\times n$ matrix on ${{\Bbb R}^+}$, $F\in C({{\Bbb R}^+, {\Bbb R}^n})$ and $\tau \in C({{\Bbb R}^+, {\Bbb R}^+})$. Using Razumikhin-type techniques and Liapunov's direct method, they establish conditions to ensure the ultimate boundedness and the global attractivity of solutions of $(*)$, and when $F(t) \equiv 0$, the asymptotic stability of the zero solution. Under those same conditions, they also show that $\int^{+\infty}_0\sum_{j=1}^{N}|B_j(t)|\,dt = +\infty$ is a necessary and sufficient condition for all of the above properties to hold. 1991 Mathematics Subject Classification: 34K15, 34C10.

Motivated by complex oriented equivariant cohomology theories, we give a natural algebraic definition of an $A$-equivariant formal group law for any abelian compact Lie group $A$. The complex orientedcohomology of the classifying space for line bundles gives an example.We also show how the choice of a complete flag gives rise to a basis and a means of calculation. This allows us to deduce thatthere is a universal ring $L_A$ for $A$-equivariant formal group laws and that it is generated by the Euler classes and the coefficients of the coproduct of the orientation. Westudy a number of topological cases in some detail. 1991 Mathematics Subject Classification: 14L05, 55N22, 55N91, 57R85.

This paper investigates the relation between a branching process and a non-linear dynamical system in $\mathbb{C}^2$. This idea has previously been fruitful in many investigations, including that of the FKPP equation by McKean, Neveu, Bramson, and others. Our concerns here are somewhat different from those in other work: we wish to elucidate those features of the dynamical system which correspond to the long-term behaviour of the random process. In particular, we are interested in how the dimension of the global attractor corresponds to that of the tail $\sigma$-algebra of the process. The Poincar\'e--Dulac operator which (locally) intertwines the non-linear system with its linearization may sometimes be exhibited as a Fourier--Laplace transform of tail-measurable random variables; but things change markedly when parameters cross values giving the `primary resonance' in the Poincar\'e--Dulac sense. Probability proves effective in establishing {\it global} properties amongst which is a clear description of the global convergence to the attractor. Several of our probabilistic results are analogues of ones obtained by Kesten and Stigum, and by Athreya and Ney, for discrete branching processes. Our simpler context allows the use of It\^o calculus. Because the paper bridges two subjects, dynamical-system theory and probability theory, we take considerable care with the exposition of both aspects. For probabilist readers, we provide a brief guide to Poincar\'e--Dulac theory; and we take the view that in a paper which we hope will be read by analysts, it would be wrong to fudge any details of rigour in our probabilistic arguments. 1991 Mathematics Subject Classification: 60H30, 60J85, 34A20.

Let $X=\operatorname{Spec} B$ be an affine varietyover a field of arbitrary characteristic, and supposethat there exists an action of a unipotent group(possibly neither smooth nor connected). Thefundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action.Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of aGalois extension, $k\hookrightarrow B=K$, which iscyclic of degree $p=\text{ch} (k)$ (that is, such thattheunipotent group is $G={\Bbb Z}/p {\Bbb Z}$),an element of minimal degree becomes anArtin--Schreier radical, and the method for computing invariants gives, in particular, theexpression for any element of $K$ in terms of theseradicals, with an explicit formula. This replacesthe well-known formula of Lagrange (which is validonly when the degree of the extension and thecharacteristic are relatively prime) in the case of an extension of degree $p=\text{ch}(k)$. (2)In this paper we give an effectiveconstruction of a stable open subset where there is a quotient. In this sense we obtain an algebraic localcriterion for the existence of a quotient in aneighbourhood. It is proved (provided the variety isnormal) that, in the following cases, such an openset is the greatest one that admits a quotient:\begin{enumerate}\item[(a)] when the action is such that the orbits havedimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any actionof the additive group $G_a$;\item[(b)] in characteristic 0, when the action is proper(obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification:primary 14L30; secondary 14D25, 14D20.

The unitary highest weight modules for $G={\rm U}(1,q)$, which occur as irreducible subrepresentations of the oscillator representation on a Fock space ${\mathcal F}$, can each be realized on a space of polynomial-valued functions over the bounded realization ${\bf B}^q$ of $G/K$. This is achieved via an integral transform constructed by L.~Mantini. A decomposition of these representations into $K$-types is given, including an explicit description of how Mantini's transform behaves on $K$-types. An inverse is produced for the transform, thus giving unitary structures for the geometric realizations of the unitary highest-weight modules over $G/K$. 1991 Mathematics Subject Classification: 22E45, 22E70.

Let $k$ be an algebraically closed field, and let $A$ be a finite-dimensional $k$-algebra. The first aim of classsical representation theory was to classify all indecomposable $A$-modules, but the homomorphisms between modules were not considered. One big step forward, made around 1975, was the invention of the Auslander--Reiten quiver of $A$. The Auslander--Reiten quiver yields a description of the category of $A$-modules modulo its infinite radical ${\rm rad}_A^\omega$. But it contains no information about the homomorphisms in ${\rm rad}_A^\omega$. We denote by mod-$A$ the category of finite-dimensional right modules over $A$. Let ${\rm rad}_A$ be the radical of mod-$A$. Following Prest, we define ${\rm rad}_A^\alpha$ for each ordinal number $\alpha$ as follows: if $J$ is an ideal in mod-$A$ and $n \geq 1$ is a natural number, then $J^n$ is the $n$-fold product of the ideal $J$. For a limit number $\beta$ define ${\rm rad}_A^\beta = \bigcap_{\alpha < \beta} {\rm rad}_A^\alpha$. Finally, let $\alpha$ be an arbitrary infinite ordinal number. Thus $\alpha = \beta + n$ for some limit number $\beta$ and some natural number $n \geq 0$. In this case, define ${\rm rad}_A^\alpha = ({\rm rad}_A^\beta)^{n+1}$. We denote the first limit number by $\omega$, the second by $\omega 2$, and so on. The minimal limit number, which is not of the form $\omega n$ for some natural number $n \geq 1$, is denoted by $\omega^2$. {\sc Theorem 1.} {\it For each ordinal number $\alpha$ with $\omega < \alpha < \omega^2$ there exists a finite-dimensional $k$-algebra $A$ with ${\rm rad}_A^\alpha \not= 0$ and ${\rm rad}_A^{\alpha + 1} = 0$. Let $Q$ be a quiver, and let $I$ be an admissible ideal of the path algebra $kQ$. We call $kQ/I$ {\it special biserial} if the following hold. Any vertex of $Q$ is the starting point of at most two arrows and also the end point of at most two arrows. Given an arrow $\beta$, there is at most one arrow $\alpha$ with $e(\alpha) = s(\beta)$ and $\alpha \beta \notin I$ and at most one arrow $\gamma$ with $e(\beta) = s(\gamma)$ and $\beta \gamma \notin I$. {\sc Theorem 2.} {\it Let $A$ be a special biserial algebra. Then the following are equivalent: \begin{enumerate} \item[{\rm (i)}] $A$ is domestic; \item[{\rm (ii)}] ${\rm rad}_A^{\omega^2} = 0$. \end{enumerate} } 1991 Mathematics Subject Classification: 16G20, 16G60, 16N40

Let $\Gamma$ be a countable abelian semigroup and $\mathcal{A}$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times \mathcal{A})$-graded Lie superalgebra $\mathfrak{L} =\bigoplus_{(\alpha, a) \Gamma \times \mathcal{A}} \mathfrak{L}_{(\alpha, a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma \times \mathcal{A})$-gradation. In this paper, we show that the Euler--Poincar\'e principle yields the generalized denominator identity for $\mathfrak{L}$ and derive a closed form formula for the supertraces $\text{str}(g| \mathcal{L}_{(\alpha, a)})$ for all $g\in G$, where $(\alpha, a) \in \Gamma \times \mathcal{A}$. We discuss the applications of our supertrace formula to various classes of infinite-dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac--Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $\text{GL}(n) \times \text{GL}(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and irreducible highest-weight modules over a generalized Kac--Moody superalgebra $\mathfrak{g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest-weight modules over the orbit Lie superalgebra $\breve{\mathfrak{g}} = \mathfrak{g}(\sigma)$ determined by $\sigma$. 1991 Mathematics Subject Classification: 17A70, 17B01, 17B65, 17B70, 11F22.

We develop a notion of Morita equivalence for general C$^{\ast}$-correspondences over C$^{\ast}$-algebras. We show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz--Pimsner algebras. Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C$^{\ast}$-algebras, then the correspondences are Morita equivalent. This generalizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure-preserving transformations. The notion of aperiodicity, which also generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated. 1991 Mathematics Subject Classification: 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.

Let $G$ be a finite group.It is proved that if the probability that two randomly chosen elements of $G$ generate a soluble group is greater than $\frac{11}{30}$ then $G$ itself is soluble.The bound is sharp, since two elements of the alternating group $A_5$ generate $A_5$ withprobability $\frac{11}{30}$.Similar probabilistic statements are proved concerning nilpotency and the property of having odd order.It is also proved that there is a number $\kappa$,strictly between $0$ and $1$, with the following property. Let $\cal X$ be any class of finite groups which is closed for subgroups, quotient groups and extensions.If the probability that two randomly chosen elementsof $G$ generate a group in $\cal X$ is greater than $\kappa$ then $G$ is in $\cal X$. The proofs use the classification of the finite simple groups and also some of the detailed information now available concerning maximal subgroups of finite almost simple groups. 1991 Mathematics Subject Classification:20F16, 20D06, 20D08, 60B99.

Let $\mathcal{B}$ be an irreducible spherical Moufang building of rank at least $2$. Thenthe group $G$ is called a group of Lie type$\mathcal{B}$ if it is generated by theroot subgroups corresponding to the rootsof some apartments of $\mathcal{B}$. This notion includes:\begin{enumerate}\item[(1)] classical groups of finite rank,\item[(2)] simple algebraic groups over arbitrary fields,\item[(3)] the `mixed' groups of Tits.\end{enumerate} General structure theorems and a general presentation type theorem for such Lie-type groups, which in a way generalize well-knowntheorems of Seitz and Curtis and Tits, are obtained. 1991 Mathematics Subject Classification:20G15, 20E42.

A semi-algebra of continuous functions is a cone $A$ of continuous real functions on a compact Hausdorff space $X$ such that $A$ contains the products of its elements. A cone $A$ is said to be of type $n$ if $f\in A$ implies $f^n(1 + f)^{-1} \in A$. Uniformly closed semi-algebras of types 0 and 1 have long been characterized in a manner analogous to the Stone--Weierstrass theorem, but, except for the case when $A$ is generated by a single function, little has been known about type 2. Here, progress is reported on two problems. The first is the characterization of those continuous linear functionals on $C(X)$ that determine semi-algebras of type 2. The second is the determination of the type of the tensor product of two type 1 semi-algebras. 1991 Mathematics Subject Classification: 46J10.