Nous étudions les polynômes $F\,\in \,\mathbb{C}\,\{{{S}_{\mathcal{T}}}\}\,\left[ Y \right]$ à coefficients dans l’anneau de germes de fonctions holomorphes au point spécial d’une variété torique affine. Nous généralisons à ce cas la paramétrisation classique des singularités quasi-ordinaires. Cela fait intervenir d’une part une généralization de l’algorithme de Newton-Puiseux, et d’autre part une relation entre le polyèdre de Newton du discriminant de $F$ par rapport à $Y$ et celui de $F$ au moyen du polytope-fibre de Billera et Sturmfels. Cela nous permet enfin de calculer, sous des hypothèses de non dégénérescence, les sommets du polyèdre de Newton du discriminant a partir de celui de $F$, et les coefficients correspondants à partir des coefficients des exposants de $F$ qui sont dans les arêtes de son polyèdre de Newton.

Let ${{\phi }_{1}},...,{{\phi }_{n}}\,(n\,\ge \,2)$ be nonzero integers such that the equation

$$\sum\limits_{i=1}^{n}{{{\phi }_{i}}x_{i}^{2}\,=\,0}$$

is solvable in integers ${{x}_{1}},...,{{x}_{n}}$ not all zero. It is shown that there exists a solution satisfying

$$0\,<\,\sum\limits_{i}^{n}{\left| {{\phi }_{i}}\left| x_{i}^{2}\,\le \,2 \right|{{\phi }_{1}}...{{\phi }_{n}}\, \right|}$$

and that the constant 2 is best possible.

We prove a relative trace formula that establishes the cubic base change for $\text{GL(2)}$. One also gets a classification of the image of base change. The case when the field extension is nonnormal gives an example where a trace formula is used to prove lifting which is not endoscopic.

For every integer $n\,>\,1$ and infinite field $F$ we construct a spectral sequence converging to the homology of $\text{G}{{\text{L}}_{n}}\left( F \right)$ relative to the group of monomial matrices $\text{G}{{\text{M}}_{n}}\left( F \right)$. Some entries in ${{E}^{2}}$-terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of $\text{G}{{\text{L}}_{n}}$ relatively to $\text{G}{{\text{L}}_{n-1}}$ and $\text{G}{{\text{M}}_{n}}$. We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.

We give a characterization of bounded plurisubharmonic functions by using their complex Monge-Ampère measures. This implies a both necessary and sufficient condition for a positive measure to be complex Monge-Ampère measure of some bounded plurisubharmonic function.

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D\,>\,3705$, there is always at least one prime $p\,<\,\sqrt{D}/2$ such that the Kronecker symbol $(D/P)\,=\,-1$.

Let ${{A}_{\theta}}$ denote the rotation algebra—the universal ${{C}^{*}}$-algebra generated by unitaries $U,V$ satisfying $VU={{e}^{2\pi i\theta }}UV$, where $\theta $ is a fixed real number. Let $\sigma $ denote the Fourier automorphism of ${{A}_{\theta}}$ defined by $U\mapsto V\text{,}V\mapsto {{U}^{-1}}$, and let ${{B}_{\theta }}={{A}_{\theta }}{{\rtimes }_{\sigma }}\mathbb{Z}/4\mathbb{Z}$ denote the associated ${{C}^{*}}$-crossed product. It is shown that there is a canonical inclusion ${{\mathbb{Z}}^{9}}\hookrightarrow {{K}_{0}}({{B}_{\theta }})$ for each $\theta $ given by nine canonical modules. The unbounded trace functionals of ${{B}_{\theta }}$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\text{H}{{\text{C}}^{0}}({{B}_{\theta }})$ when $\theta $ is irrational. The Chern characters of the nine modules—and more importantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi’s theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense ${{\text{G}}_{\delta }}$ set of values of $\theta $ one has ${{K}_{0}}({{B}_{\theta }})\cong {{\mathbb{Z}}^{9}}$ and is generated by the nine classes constructed here.

To each field $F$ of characteristic not 2, one can associate a certain Galois group ${{\mathcal{G}}_{F}}$, the so-called $\text{W}$-group of $F$, which carries essentially the same information as the Witt ring $W(F)$ of $F$. In this paper we investigate the connection between ${{\mathcal{G}}_{F}}$ and ${{\mathcal{G}}_{F(\sqrt{a})}}$, where $F(\sqrt{a})$ is a proper quadratic extension of $F$. We obtain a precise description in the case when $F$ is a pythagorean formally real field and $a=-1$, and show that the $\text{W}$-group of a proper field extension $K/F$ is a subgroup of the $\text{W}$-group of $F$ if and only if $F$ is a formally real pythagorean field and $K=F(\sqrt{-1)}$. This theorem can be viewed as an analogue of the classical Artin-Schreier’s theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when $a$ is a double-rigid element in $F$. Some of these results carry over to the general setting.

Let ${{G}_{n}}$ be the split classical groups $\text{Sp}(\text{2}n\text{),}\,\text{SO(2}n\text{+1})$ and $\text{SO(2}n\text{)}$ defined over a $p$-adic field F or the quasi-split classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a quadratic extension $E/F$. We prove the self-duality of unitary supercuspidal data of standard Levi subgroups of ${{G}_{n}}(F)$ which give discrete series representations of ${{G}_{n}}(F)$.

Subadditivity, sublinearity, submultiplicativity, and other conditions are considered for spectra of pairs of operators on a Hilbert space. Sublinearity, for example, is a weakening of the well-known property $L$ and means $\sigma (A\,+\,\lambda B)\,\subseteq \,\sigma (A)\,+\,\lambda \sigma (B)$ for all scalars $\lambda$. The effect of these conditions is examined on commutativity, reducibility, and triangularizability of multiplicative semigroups of operators. A sample result is that sublinearity of spectra implies simultaneous triangularizability for a semigroup of compact operators.

We study estimates of the type

1

$${{\left\| \phi (D)\,-\,\phi ({{D}_{0}}) \right\|}_{E(\mathcal{M},\,\tau )}}\,\le \,C\,\cdot \,{{\left\| D\,-\,{{D}_{0}} \right\|}^{\alpha }},\,\,\,\,\,\,\,\alpha \,=\,\frac{1}{2},\,1$$

where $\phi (t)\,=\,t{{(1\,+\,{{t}^{2}})}^{-1/2}},\,{{D}_{0}}\,=\,D_{0}^{*}$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $M,D-{{D}_{0}}$ is a bounded self-adjoint linear operator from $\mathcal{M}$ and ${{(1+D_{0}^{2})}^{-1/2}}\in E(M,\tau )$, where $E(\mathcal{M},\tau )$ is a symmetric operator space associated with $\mathcal{M}$. In particular, we prove that $\phi \left( D \right)-\phi \left( {{D}_{0}} \right)$ belongs to the non-commutative ${{L}_{p}}$-space for some $p\in (1,\infty )$, provided ${{(1+D_{0}^{2})}^{-1/2}}$ belongs to the noncommutative weak ${{L}_{r}}$-space for some $r\in [1,p)$. In the case $\mathcal{M}\,=\,\mathcal{B}\left( \mathcal{H} \right)$ and $1\,\le \,p\,\le \,2$, we show that this result continues to hold under the weaker assumption ${{(1+D_{0}^{2})}^{-1/2}}\in {{C}_{p}}$. This may be regarded as an odd counterpart of A. Connes’ result for the case of even Fredholm modules.

${{H}^{p}}$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

The main new results in this paper are contained in the geometric Theorems 1 and 2 of Section 0.1 below and they are related to previous results of M. Gromov and of myself (cf. [11], [29]). These results are used to prove some general potential theoretic estimates on Lie groups (cf. Section 0.3) that are related to my previous work in the area (cf. [28], [34]) and to some deep recent work of G. Alexopoulos (cf. [3], [4]).

In the first part of this paper generalizations of Hesselink’s $q$-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a $q$-analog of the Kostant-Rallis theorem is given for the real group $\text{SL(4,}\,\mathbb{R}\text{)}$ (that is $\text{SO}(4)$ acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.