944 research outputs found
A construction of polynomials with squarefree discriminants
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we
give an unconditional construction of infinitely many monic irreducible
polynomials of degree n with integer coefficients having squarefree
discriminant and exactly r real roots. These give rise to number fields of
degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we
may also force the discriminant to be coprime to any given integer. The number
of fields produced with discriminant in the range [-N, N] is at least c
N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic
number fields admit everywhere unramified degree n extensions whose normal
closures have Galois group A_n. This generalizes results of Yamamura, who
treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not
control the real place.Comment: 10 pages; v2: refereed version, minor edits onl
Level sets and non Gaussian integrals of positively homogeneous functions
We investigate various properties of the sublevel set
and the integration of on this sublevel set when and are positively
homogeneous functions. For instance, the latter integral reduces to integrating
on the whole space (a non Gaussian integral) and when is
a polynomial, then the volume of the sublevel set is a convex function of the
coefficients of . In fact, whenever is nonnegative, the functional is a convex function of for a large class of functions
. We also provide a numerical approximation scheme to compute
the volume or integrate (or, equivalently to approximate the associated non
Gaussian integral). We also show that finding the sublevel set of minimum volume that contains some given subset is a
(hard) convex optimization problem for which we also propose two convergent
numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian
integrals for homogeneous polynomials that are sums of squares and critical
points of a specific function
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