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1. CJM 2009 (vol 61 pp. 205)
Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability $1$. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.
Categories:13J30, 12Y05, 13P99, 14P10, 90C22 |
2. CJM 2002 (vol 54 pp. 897)
The Valuative Theory of Foliations This paper gives a characterization of valuations that follow the
singular infinitely near points of plane vector fields, using the
notion of L'H\^opital valuation, which generalizes a well known classical
condition. With that tool, we give a valuative description of vector
fields with infinite solutions, singularities with rational quotient
of eigenvalues in its linear part, and polynomial vector fields with
transcendental solutions, among other results.
Categories:12J20, 13F30, 16W60, 37F75, 34M25 |
3. CJM 2001 (vol 53 pp. 33)
Merit Factors of Polynomials Formed by Jacobi Symbols We give explicit formulas for the $L_4$ norm (or equivalently for the
merit factors) of various sequences of polynomials related to the
polynomials
$$
f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n.
$$
and
$$
f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n.
$$
where $(\frac{\cdot}{N})$ is the Jacobi symbol.
Two cases of particular interest are when $N = pq$ is a product of two
primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt,
Jensen and Jensen and of the authors.
This study arises from a number of conjectures of Erd\H{o}s,
Littlewood and others that concern the norms of polynomials with
$-1,1$ coefficients on the disc. The current best examples are of the
above form when $N$ is prime and it is natural to see what happens for
composite~$N$.
Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols Categories:11J54, 11B83, 12-04 |
4. CJM 2000 (vol 52 pp. 1018)
Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$-equivariant centers, into a $G$-variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.
Categories:14L30, 14E15, 14E05, 12E05, 20G10 |
5. CJM 2000 (vol 52 pp. 833)
W-Groups under Quadratic Extensions of Fields To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the so-called 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 $\wg$ and
$\G_{F(\sqrt{a})}$, 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 W-group of a proper field extension $K/F$ is a subgroup of the
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.
Categories:11E81, 12D15 |
6. CJM 1999 (vol 51 pp. 69)
On a Theorem of Hermite and Joubert A classical theorem of Hermite and Joubert asserts that any field
extension of degree $n=5$ or $6$ is generated by an element whose
minimal polynomial is of the form $\lambda^n + c_1 \lambda^{n-1} +
\cdots + c_{n-1} \lambda + c_n$ with $c_1=c_3=0$. We show that this
theorem fails for $n=3^m$ or $3^m + 3^l$ (and more generally, for $n =
p^m$ or $p^m + p^l$, if 3 is replaced by another prime $p$), where $m
> l \geq 0$. We also prove a similar result for division algebras and
use it to study the structure of the universal division algebra $\UD
(n)$.
We also prove a similar result for division algebras and use it to
study the structure of the universal division algebra $\UD(n)$.
Categories:12E05, 16K20 |
7. CJM 1999 (vol 51 pp. 10)
Tractable Fields A field $F$ is said to be tractable when a condition
described below on the simultaneous representation of
quaternion algebras holds over $F$. It is shown
that a global field $F$ is tractable i{f}f $F$ has
at most one dyadic place. Several other examples
of tractable and nontractable fields are given.
Categories:12E15, 11R52 |
8. CJM 1998 (vol 50 pp. 1189)
Totally real rigid elements and Galois theory Abelian closed subgroups of the Galois group of the pythagorean closure
of a formally real field are described by means of the inertia group of
suitable valuation rings.
Categories:12F10, 12J20 |