1,490 research outputs found
Real root finding for equivariant semi-algebraic systems
Let be a real closed field. We consider basic semi-algebraic sets defined
by -variate equations/inequalities of symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by .
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by polynomials of degree
in time . This improves the state-of-the-art which is exponential
in . When the variables are quantified and the
coefficients of the input system depend on parameters , one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time
Real root finding for equivariant semi-algebraic systems
International audienceLet be a real closed field. We consider basic semi-algebraic sets defined by -variate equations/inequalities of symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by . Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by polynomials of degree in time . This improves the state-of-the-art which is exponential in . When the variables are quantified and the coefficients of the input system depend on parameters , one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time
Branching laws for Verma modules and applications in parabolic geometry. I
We initiate a new study of differential operators with symmetries and combine
this with the study of branching laws for Verma modules of reductive Lie
algebras. By the criterion for discretely decomposable and multiplicity-free
restrictions of generalized Verma modules [T. Kobayashi,
http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we are
brought to natural settings of parabolic geometries for which there exist
unique equivariant differential operators to submanifolds. Then we apply a new
method (F-method) relying on the Fourier transform to find singular vectors in
generalized Verma modules, which significantly simplifies and generalizes many
preceding works. In certain cases, it also determines the Jordan--H\"older
series of the restriction for singular parameters. The F-method yields an
explicit formula of such unique operators, for example, giving an intrinsic and
new proof of Juhl's conformally invariant differential operators [Juhl,
http://dx.doi.org/10.1007/978-3-7643-9900-9 {Progr. Math. 2009}] and its
generalizations. This article is the first in the series, and the next ones
include their extension to curved cases together with more applications of the
F-method to various settings in parabolic geometries
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