Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConstruction of Identities for Sums of Squares
56
0
0.0
(
0
)
Authors
Daniel B. Shapiro
Ask ChatGPT about the research
No Arabic abstract
From sum-of-squares formulas of sizes $[r, s, n]$ and $[r, s, n]$ we construct a formula of size $[r + r, 2ss, 2nn]$.
rate research
Read More
We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-SzegH{o} theorem can be used to simplify the problem. When the Young subgroup has only two orbits, such a matroid is strongly Rayleigh if and only if an associated univariate polynomial has only real roots. When this polynomial is quadratic we get an explicit structural criterion for the strong Rayleigh property. Finally, if one of the orbits has rank two then the matroid is strongly Rayleigh if and only if the Rayleigh difference of any two points on this line is in fact a sum of squares.
The Macaulay2 package SumsOfSquares decomposes polynomials as sums of squares. It is based on methods to rationalize sum-of-squares decompositions due to Parrilo and Peyrl. The package features a data type for sums-of-squares polynomials, support for external semidefinite programming solvers, and optimization over varieties.
The Hilberts 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, as to the question whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials associated with isoparametric polynomials (initiated by E. Cartan) which define the focal submanifolds of the corresponding isoparametric hypersurfaces.
When the sequences of squares of primes is coloured with $K$ colours, where $K geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of the same colour. We show that $s(K) ll K expleft(frac{(3log 2 + {rm o}(1))log K}{log log K}right)$ for $K geq 2$. This improves on $s(K) ll_{epsilon} K^{2 +epsilon}$, which is the best available upper bound for $s(K)$.
We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P^3 whose function field has level 2 is dense in the set of those that have no real points.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
