55 research outputs found
On the degree and half degree principle for symmetric polynomials
In this note we aim to give a new, elementary proof of a statement that was
first proved by Timofte. It says that a symmetric real polynomial of degree
in variables is positive on (on ) if and only
if it is so on the subset of points with at most
distinct components. We deduce Timofte's original statement as a corollary of a
slightly more general statement on symmetric optimization problems. The idea we
are using to prove this statement is to relate it to a linear optimization
problem in the orbit space. The fact that for the case of the symmetric group
this can be viewed as a question on normalized univariate real
polynomials with only real roots allows us to conclude the theorems in a very
elementary way. We hope that the methods presented here will make it possible
to derive similar statements also in the case of other groups.Comment: (v2) revision based on suggestions by refere
Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets
Let be a real closed field and
an ordered domain. We consider the algorithmic problem of computing the
generalized Euler-Poincar\'e characteristic of real algebraic as well as
semi-algebraic subsets of , which are defined by symmetric
polynomials with coefficients in . We give algorithms for computing
the generalized Euler-Poincar\'e characteristic of such sets, whose
complexities measured by the number the number of arithmetic operations in
, are polynomially bounded in terms of and the number of
polynomials in the input, assuming that the degrees of the input polynomials
are bounded by a constant. This is in contrast to the best complexity of the
known algorithms for the same problems in the non-symmetric situation, which
are singly exponential. This singly exponential complexity for the latter
problem is unlikely to be improved because of hardness result
(-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with
arXiv:1312.658
Symmetric nonnegative forms and sums of squares
We study symmetric nonnegative forms and their relationship with symmetric
sums of squares. For a fixed number of variables and degree , symmetric
nonnegative forms and symmetric sums of squares form closed, convex cones in
the vector space of -variate symmetric forms of degree . Using
representation theory of the symmetric group we characterize both cones in a
uniform way. Further, we investigate the asymptotic behavior when the degree
is fixed and the number of variables grows. Here, we show that, in
sharp contrast to the general case, the difference between symmetric
nonnegative forms and sums of squares does not grow arbitrarily large for any
fixed degree . We consider the case of symmetric quartic forms in more
detail and give a complete characterization of quartic symmetric sums of
squares. Furthermore, we show that in degree the cones of nonnegative
symmetric forms and symmetric sums of squares approach the same limit, thus
these two cones asymptotically become closer as the number of variables grows.
We conjecture that this is true in arbitrary degree .Comment: (v4) minor revision and small reorganizatio
Mathematische Unterhaltung nicht nur an Sonntagen : George Szpiros BĂŒcher sind ein lehrreiches VergnĂŒgen
Rezension zu: George G. Szpiro : Mathematik fĂŒr Sonntagmorgen : 50 Geschichten aus Mathematik und Wissenschaft, NZZ Verlag, ZĂŒrich 2006, ISBN 978-3-03823-353-4 ; 240 Seiten, 26 Euro/38 CHF George G. Szpiro : Mathematik fĂŒr Sonntagnachmittag : Weitere 50 Geschichten aus Mathematik und Wissenschaft, NZZ Verlag, ZĂŒrich 2006, ISBN 978-3-03823-225-4 ; 236 Seiten, 26 Euro/38 CH
Reflection groups and cones of sums of squares
We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the An, Bn and Dn case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2d stabilizes with. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms
A Note on Extrema of Linear Combinations of Elementary Symmetric Functions
This note provides a new approach to a result of Foregger and related earlier
results by Keilson and Eberlein. Using quite different techniques, we prove a
more general result from which the others follow easily. Finally, we argue that
the proof given by Foregger is flawed.Comment: (v2) revision based on suggestions by refere
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