3,534 research outputs found
Real radical initial ideals
We explore the consequences of an ideal I of real polynomials having a real
radical initial ideal, both for the geometry of the real variety of I and as an
application to sums of squares representations of polynomials. We show that if
in_w(I) is real radical for a vector w in the tropical variety, then w is in
the logarithmic set of the real variety. We also give algebraic sufficient
conditions for w to be in the logarithmic limit set of a more general
semialgebraic set. If in addition the entries of w are positive, then the
corresponding quadratic module is stable. In particular, if in_w(I) is real
radical for some positive vector w then the set of sums of squares modulo I is
stable. This provides a method for checking the conditions for stability given
by Powers and Scheiderer.Comment: 16 pages, added examples, minor revision
Real Algebraic Geometry With A View Toward Systems Control and Free Positivity
New interactions between real algebraic geometry, convex optimization and free non-commutative geometry have recently emerged, and have been the subject of numerous international meetings. The aim of the workshop was to bring together experts, as well as young researchers, to investigate current key questions at the interface of these fields, and to explore emerging interdisciplinary applications
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
We deploy algebraic complexity theoretic techniques for constructing
symmetric determinantal representations of for00504925mulas and weakly skew
circuits. Our representations produce matrices of much smaller dimensions than
those given in the convex geometry literature when applied to polynomials
having a concise representation (as a sum of monomials, or more generally as an
arithmetic formula or a weakly skew circuit). These representations are valid
in any field of characteristic different from 2. In characteristic 2 we are led
to an almost complete solution to a question of B\"urgisser on the
VNP-completeness of the partial permanent. In particular, we show that the
partial permanent cannot be VNP-complete in a finite field of characteristic 2
unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on
Randomization, Relaxation, and Complexity in Polynomial Equation Solving,
edited by Gurvits, Pebay, Rojas and Thompso
- …