# Formas semidefinidas positivas e somas de quadrados

### Full-text

Carla Fidalgo, Jul 05, 2015 Available from:- References (7)
- Cited In (0)

- [Show abstract] [Hide abstract]

**ABSTRACT:**A. Hurwitz [ibid. 108, 266-268 (1891); Coll. Works II (Basel 1933; Zbl 0007.19504), 505-507] proved the arithmetic-geometric inequality by writing the form F n (x 1 ,···,x n )=x 1 n +···+x n n -nx 1 ···x n as a sum of non-negative terms. A key identity involved writing the form F 2d as a sum of squares of forms. Hurwitz’ construction was explicit and used a large number of squares - (1+d 2 (d-1)2 d-3 ). We give an alternate, algorithmic construction which requires fewer (≈6d) squares, and applies to a somewhat larger class of forms related to the arithmetic-geometric inequality. The ”length” of a form f is the smallest T for which there exist real forms h 1 ,···,h T with f=∑h k 2 . Let ℋ(2d) denote the length of F 2d . We show that ℋ(2d)≤6d-7 for d≥2, ℋ(2 k )≤2 k -1, with equality for k=1 and 2, ℋ(6)≤9 and ℋ(4rd)≤2rℋ(2d)+ℋ(2r). - [Show abstract] [Hide abstract]

**ABSTRACT:**By a diagonal minus tail form (of even degree) we understand a real homogeneous polynomial F(x 1, . . . , x n ) = F(x) = D(x) − T(x), where the diagonal part D(x) is a sum of terms of the form bi xi2d{b_i x_i^{2d}} with all b i ≥ 0 and the tail T(x) a sum of terms ai1i2¼inx1i1¼xnin{a_{i_1i_2\cdots i_n}x_1^{i_1}\cdots x_n^{i_n}} with ${a_{i_1i_2\cdots i_n} > 0}${a_{i_1i_2\cdots i_n} > 0} and at least two i ν ≥ 1. We show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials. We also give an easily tested sufficient condition for a polynomial to be a sum of squares of polynomials (sos) and show that the class of polynomials passing this test is wider than the class passing Lasserre’s recent conditions. Another sufficient condition for a polynomial to be sos, like Lasserre’s piecewise linear in its coefficients, is also given.Mathematische Zeitschrift 01/2009; 269(3):629-645. DOI:10.1007/s00209-010-0753-y - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for the volumes of the bases. If the degree is greater than 2 then we show that the ratio of the volumes of the bases, raised to the power reciprocal to the ambient dimension, tends to 0 as the number of variables tends to infinity.