Sums of squares, moment matrices and optimization over polynomials

In IMA Vol 149 Emerging Applications of Algebraic Geometry, Springer, pages 157-270, 2009.

• In equation (1.12), the objective function should read < C,X > (replace A by X).

• In Example 2.10 (continued), p. 177, the kernel of $M_y^T$ consists of the vectors $u$ satisfying $u_2 = u_3 = u_5 =0$.

• In line 3 of the proof of Lemma 3.3, replace $p(x)$ by $p(x/x_{n+1})$.

• In line 4 of the proof of Lemma 3.6 (top of page 180), $c$ and its transpose should be interchanged in matrix $\tilde Q$.

• In the second line after Theorem 3.11, the identity should read: $1 = h_1 (1-x)/2 + h_2 (1/2)$.

• In the line above equation (3.12), $g_0$ stands for $g_J$ when $J$ is the empty set.

• In Theorem 3.20, modume ==> module.

• In Lemma 4.2 (ii), the defined set is $K$ (used in the proof).

• At line 4 of the proof of Lemma 5.15: in the definition of $\delta_{..,..}$, replace the second term $\beta-\beta'$ by $\alpha-\beta'$.

• The second proof of Theorem 5.14 (Flat extension theorem) given on pages 210-211 (following Schweighofer [133]) has a flaw; namely, it is not true that the set $U$ in Lemma 5.18 is a linear space.

• In equation (6.7): replace $L(f)$ by $L(p)$ in the objective function of the minimization program.

• In the second paragraph of the proof of Lemma 7.21, the definition for the set $W_0$ in the displayed equation is not correct as it is now; indeed it could be that some $W_i$ ($i>=1$) contains no real point, in which case one cannot claim later that the value $a_i$ is nonnegative/positive.
This can be fixed as follows: Define the set $W_0$ as the union of all the irreducible components $V_l$ of the gradient variety for which there does not exist any other irreducible component $V_{l'}$ with $p(V_l)=p(V_{l'})$ and $V_{l'}$ contains at least one real point. Then $W_0$ contains no real point (needed in the proof of Lemma 7.22) and any other set $W_i$ does contain a real point.

Thanks in advance for mailing me further corrections.