Corrections for the IMA survey (original) (raw)

This page contains corrections to the survey article (posted version):

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 MyTM_y^TMyT consists of the vectors uuu satisfying u2=u3=u_5=0u_2 = u_3 = u_5 =0u_2=u_3=u_5=0.

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

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

• In the second line after Theorem 3.11, the identity should read: 1=h1(1−x)/2+h2(1/2)1 = h_1 (1-x)/2 + h_2 (1/2)1=h_1(1−x)/2+h_2(1/2).

• In the line above equation (3.12), g0g_0g0 stands for gJg_JgJ when JJJ is the empty set.

• In Theorem 3.20, modume ==> module.

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

• At line 4 of the proof of Lemma 5.15: in the definition of delta..,..\delta_{..,..}delta..,.., replace the second term beta−beta′\beta-\beta'beta−beta′ by alpha−beta′\alpha-\beta'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 UUU in Lemma 5.18 is a linear space.

• In equation (6.7): replace L(f)L(f)L(f) by L(p)L(p)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 W0W_0W0 in the displayed equation is not correct as it is now; indeed it could be that some WiW_iWi ($i>=1$) contains no real point, in which case one cannot claim later that the value aia_iai is nonnegative/positive.
  This can be fixed as follows: Define the set W0W_0W0 as the union of all the irreducible components VlV_lVl of the gradient variety for which there does not exist any other irreducible component Vl′V_{l'}Vl′ with p(Vl)=p(Vl′)p(V_l)=p(V_{l'})p(Vl)=p(Vl′) and Vl′V_{l'}Vl′ contains at least one real point. Then W0W_0W0 contains no real point (needed in the proof of Lemma 7.22) and any other set WiW_iWi does contain a real point.

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