61 research outputs found
Positive Gorenstein Ideals
We introduce positive Gorenstein ideals. These are Gorenstein ideals in the
graded ring \RR[x] with socle in degree 2d, which when viewed as a linear
functional on \RR[x]_{2d} is nonnegative on squares. Equivalently, positive
Gorenstein ideals are apolar ideals of forms whose differential operator is
nonnegative on squares. Positive Gorenstein ideals arise naturally in the
context of nonnegative polynomials and sums of squares, and they provide a
powerful framework for studying concrete aspects of sums of squares
representations. We present applications of positive Gorenstein ideals in real
algebraic geometry, analysis and optimization. In particular, we present a
simple proof of Hilbert's nearly forgotten result on representations of ternary
nonnegative forms as sums of squares of rational functions. Drawing on our
previous work, our main tools are Cayley-Bacharach duality and elementary
convex geometry
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
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