957 research outputs found
On Projective Equivalence of Univariate Polynomial Subspaces
We pose and solve the equivalence problem for subspaces of ,
the dimensional vector space of univariate polynomials of degree . The group of interest is acting by projective transformations
on the Grassmannian variety of -dimensional
subspaces. We establish the equivariance of the Wronski map and use this map to
reduce the subspace equivalence problem to the equivalence problem for binary
forms
Reduced Gr\"obner Bases of Certain Toric Varieties; A New Short Proof
Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of
positive integers. Let C be a toric variety in the affine (n+1)-space, defined
parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a
minimal Gr\"obner basis for the toric ideal which is the defining ideal of C
and give sufficient and necessary conditions for this basis to be the reduced
Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a
much simpler proof than that of \cite{Ayy}
The Geometry of Fixed Point Varieties on Affine Flag Manifolds
Let be a semisimple, simply connected, algebraic group over an
algebraically closed field with Lie algebra . We study the spaces
of parahoric subalgebras of a given type containing a fixed nil-elliptic
element of , i.e. fixed point varieties on affine flag
manifolds. We define a natural class of -actions on affine flag manifolds,
generalizing actions introduced by Lusztig and Smelt. We formulate a condition
on a pair consisting of and a
-action of the specified type which guarantees that induces an
action on the variety of parahoric subalgebras containing .
For the special linear and symplectic groups, we characterize all regular
semisimple and nil-elliptic conjugacy classes containing a representative whose
fixed point variety admits such an action. We then use these actions to find
simple formulas for the Euler characteristics of those varieties for which the
-fixed points are finite. We also obtain a combinatorial description of
the Euler characteristics of the spaces of parabolic subalgebras containing a
given element of certain nilpotent conjugacy classes of .Comment: Latex2e, 33 pages. To appear in Transactions of the AM
Higher uniformity of bounded multiplicative functions in short intervals on average
Let denote the Liouville function. We show that, as ,
for all fixed and with fixed
but arbitrarily small. Previously this was only established for . We
obtain this result as a special case of the corresponding statement for
(non-pretentious) -bounded multiplicative functions that we prove. In fact,
we are able to replace the polynomial phases by degree
nilsequences . By the inverse theory for the Gowers
norms this implies the higher order asymptotic uniformity result
in the same
range of . We present applications of this result to patterns of various
types in the Liouville sequence. Firstly, we show that the number of sign
patterns of the Liouville function is superpolynomial, making progress on a
conjecture of Sarnak about the Liouville sequence having positive entropy.
Secondly, we obtain cancellation in averages of over short polynomial
progressions , which in the case of linear
polynomials yields a new averaged version of Chowla's conjecture. We are in
fact able to prove our results on polynomial phases in the wider range , thus strengthening also previous work on the
Fourier uniformity of the Liouville function.Comment: 104 page
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