We prove that there are two incomplete d.r.e.\ degrees (the Turing degrees of differences of two recursively enumerable sets) such that every non-zero recursively enumerable degree cups at least one of them to ${\bf 0}′$, the greatest recursively enumerable (Turing) degree.

We consider the problem of minimizing the uniform norm on $[0, 1]$ over non-zero polynomials $p$ of the form

$$p(x) = \sum_{j=0}^n a_jx^j \quad\text{with } |a_j| \le 1,\, a_j \in {\Bbb C},$$

where the modulus of the first non-zero coefficient is at least $\delta > 0$. Essentially sharp bounds are given for this problem. An interesting related result states that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that

$$ \exp (-c_1 \sqrt{n} ) \le \inf_{0 \ne p \in {\cal F}_n }\|p\|_{[0, 1]} \le \exp (-c_2 \sqrt{n}),$$

for every $n \ge 2$, where ${\cal F}_n$ denotes the set of polynomials of degree at most $n$ with coefficients from $\{-1, 0, 1 \}$. This Chebyshev-type problem is closely related to the question of how many zeros a polynomial from the above classes can have at $1$. We also give essentially sharp bounds for this problem.

{\em Inter alia} we prove that there is an absolute constant $c > 0$ such that every polynomial $p$ of the form

$$p(x) = \sum_{j=0}^n a_j x^j, \quad\text{with } |a_j| \le 1,\,|a_0|=|a_n|=1,\, a_j \in {\Bbb C},$$

has at most $c\sqrt{n}$ real zeros. This improves the old bound $c\sqrt{n\log n}$ given by Schur in 1933, as well as more recent related bounds of Bombieri and Vaaler, and, up to the constant $c$, this is the best possible result.

All the analysis rests critically on the key estimate stating that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that

$$|f(0)|^{c_1/a} \le \exp (c_2/a)\|f\|_{[1-a,1]},$$

for every $f \in {\cal S}$ and $a \in (0, 1]$, where $\cal S$ denotes the collection of all analytic functions $f$ on the open unit disk $D := \{ z \in {\Bbb C}: |z| < 1 \}$ that satisfy

$$|f(z)| \le \frac{1}{1-|z|}\quad \text{for } z \in D.$$

Suppose given a finite Galois extension $K/k$ of number fields with group $G$ and a finite sufficiently large $G$-stable set $S$ of primes of $K$, write $E$ for the group of $S$-units in $K$, and $\Delta S$ for the augmentation submodule of the $\Bbb{Z}$-free permutation module $\Bbb{Z} S$ on $S$. Let also $A$, $B$ be finitely generated cohomologically trivial $\Bbb{Z} G$-modules, and $\tau$ Tate's canonical class defining a 2-extension $E\rightarrowtail A\to B\twoheadrightarrow\Delta S$. Chinburg has assigned the algebraic invariant $\Omega=[A]-[B]\in K_0(\Bbb{Z}G)$ to $\tau$ and conjectured that $\Omega$ equals the root number class $W(K/k)$, an analytic invariant defined by Cassou-Nogués and Fröhlich, and based on Artin's root numbers. We combine $\Omega$ with $G$-injections $\varphi:\Delta S\rightarrowtail E$ to obtain lifts $\Omega_\varphi\in K_0T(\Bbb{Z}G)$ of $\Omega$ in the Grothendieck group of finite $\Bbb{Z}G$-modules of finite projective dimension. Unlike $K_0(\Bbb{Z}G)$, the group $K_0T(\Bbb{Z}G)$ has a natural decomposition as a direct sum $\bigoplus K_0T(\Bbb{Z}_lG)$, over all finite rational primes $l$. This provides a local setting for the study of the $\Omega_\varphi$.

Each of the above $G$-equivariant maps $\varphi$ determines a function $A_\varphi$ on characters of $G$ via $L$-values at zero. Stark's conjecture asserts that $A_\varphi$ is Galois-equivariant. These functions can be viewed as naming elements of $K_0T(\Bbb{Z}G)$ by using Fröhlich's Hom description. The new conjecture, referred to as the Lifted Root Number Conjecture, is that $A_\varphi$ names $\Omega_\varphi$. The truth of this conjecture would imply that of Chinburg's.

The dependence of the invariant $\Omega_\varphi$ on $S$, how it varies with $\varphi$, and its behaviour under restriction to subgroups and deflation to factor groups of $G$, are studied in individual sections. Then, the Hom description is used to compare $\Omega_\varphi$ with $A_\varphi$ whenever Stark's conjecture holds true. It is shown that it suffices to give a proof of the Lifted Root Number Conjecture at the prime $l$ in the case when the Galois group $G$ is replaced by an $l$-elementary group, that is, the direct product of an $l$-group and a cyclic group of order prime to $l$.

We construct the moduli spaces $M(n)$ of semistable parabolic sheaves of rank $n$ and fixed Euler characteristic on a disjoint union $X$ of integral projective curves with parabolic structures over Cartier divisors on $X$. In the case where $X$ is non-singular, $M$ is a normal projective variety. Suppose that $X$ is the desingularisation of a reducible reduced curve $Y$ with at most ordinary double points as singularities. We show that, for a suitable choice of parabolic structure, $M(n)$ is the normalisation of the moduli space of torsion-free sheaves of rank $n$ and fixed Euler characteristic on $Y$, and it is a desingularisation if semistability coincides with stability. We find explicit descriptions of $M(n)$ for small $n$ in some cases.

A function $f$ from the symmetric group $S_n$ into $\Bbb R$ is called a class function if $f(\sigma ) = f(\tau )$ whenever $\tau$ is conjugate to $\sigma$. Let $d_f$ be thegeneralized matrix function associated with $f$, mapping the $n$-by-$n$ positive semidefinite Hermitian matricesto $\Bbb R$. For example, if $f(\sigma ) = \mbox{sgn}(\sigma )$, then $d_f(A) = \det A$.

We consider the cone $K_n$ of those $f$ for which $d_f(A) \geq 0$ for all $n$-by-$n$ positive semidefinite Hermitian matrices. For $n = 4$ we show that $K_n$ is polyhedral, and explicitly find the extreme rays. Equivalently, $f$ belongs to $K_n$ if $d_f(A) \geq 0$ for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order $n = 4$.

We obtain similar results for 4-by-4 real positive semidefinite matrices.

There is a standard additive decomposition of the Hochschild cohomology ring of the group algebra of a finite group $G$ as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of $G$. A special case of our main result describes the cup product in terms of this decomposition. As applications, we determine presentations for the Hochschild cohomology rings of

[(1)] the mod-$3$ group algebra of the symmetric group $S_3$,

[(2)] the mod-$2$ group algebra of the alternating group $A_4$, and

[(3)] the mod-$2$ group algebras of the dihedral $2$-groups.

Let $V$ be a vector space and let $\{ e_1,\hdots,e_r \}$ be a basis of $V$. An algebra structure on $V$ is given by $r^3$ structure constants $c_{ij}^h$ where $e_i\cdot e_j = \sum_h c_{ij}^h e_h$. We require this algebra structure to be associative with unit element $e_1$. This limits the sets of structure constants $(c_{ij}^h)$ to a subvariety of $k^{r^3}$, which we denote by $\mbox{Alg}_r$. Base changes in $V$ (leaving $e_1$ fixed) give rise to the natural transport of structure action on $\mbox{Alg}_r$; isomorphism classes of $r$-dimensional algebras are in one-to-one correspondence with the orbits under this action.

In this paper we classify the smooth closed subvarieties of $\mbox{Alg}_r$ which are invariant under the transport of structure action and study the singularities which may occur. In particular, we show that if $r=n^2$ then the closure of the locus corresponding to the matrix algebra $M_n(k)$ is not smooth for $n \geq 3$. This gives a negative answer to a question of Seshadri on the desingularization of moduli spaces of vector bundles over curves.

We study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets, typical examples being the Meixner polynomials and the Charlier polynomials. Following ideas of Rakhmanov, Dragnev and Saff, weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials, which can be solved explicitly. We give formulas for the contracted zero distributions, the $n$th root asymptotics and the asymptotics of the largest zeros.

Denote by $\Bbb A$ the class of all absolutely continuous contractions whose associated Sz. Nagy-Foias functional calculus is isometric. Starting from the fact that if $u$ is a non-constant inner function and if $T\in {\Bbb A}$, then so does $u(T)$, we study how inner functions operate on the classes ${\Bbb A}_{n,m}$, subclasses of the class ${\Bbb A}$. For this purpose, we use standard dual algebra techniques and a decomposition of the algebra $H^\infty$ into a weak*-topological direct sum of copies of itself. We also discuss mapping theorems for the support of the spectral measures associated with the unitary parts of the minimal isometric extension and the minimal co-isometric extension of an absolutely continuous contraction $T$.