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Algebraic Geometry and Sums-Of-Squares

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Algebraic Geometry and Sums-Of-Squares Algebraic geometry and sums-of-squares. Mauricio Velasco (Universidad de los Andes, Colombia) Joint Mathematics Meetings 2019 AMS Short Course (Lecture 1) Motivation Let f 2 R[X1;:::; Xn] be a multivariate polynomial with real coefficients. Definition. n The polynomial f is nonnegative if f (α) ≥ 0 for every α 2 R Definition. The polynomial f is a sum-of-squares if there exist an integer t > 0 and polynomials g1;:::; gt 2 R[X1;:::; Xn] such that 2 2 f = g1 + ··· + gt : Nonnegative polynomials Definition. The global polynomial optimization problem asks us to find α∗ := inf f (α) α2Rn Nonnegative polynomials Definition. The global polynomial optimization problem asks us to find α∗ := inf f (α) α2Rn Remark. ∗ α = supfλ 2 R : f (x) − λ ≥ 0g Nonnegative polynomials Definition. The global polynomial optimization problem asks us to find α∗ := inf f (α) α2Rn Remark. ∗ α = supfλ 2 R : f (x) − λ ≥ 0g So we can reformulate polynomial optimization problems as linear optimization over some affine slice of the cone of nonnegative polynomials of degree ≤ deg(f ). Sums-of-squares Let V ⊆ R[X0;:::; Xn] be a vector subspace of polynomials of dimension e. Lemma. A polynomial f (x) is a sum-of-squares of elements of V if and only e×e if there exists a symmetric matrix A 2 R such that A 0 and f (x) = m~ t Am~ t where m~ = (h1;:::; he ) is a vector whose entries are a basis for V . Sums-of-squares Let V ⊆ R[X0;:::; Xn] be a vector subspace of polynomials of dimension e. Lemma. A polynomial f (x) is a sum-of-squares of elements of V if and only e×e if there exists a symmetric matrix A 2 R such that A 0 and f (x) = m~ t Am~ t where m~ = (h1;:::; he ) is a vector whose entries are a basis for V . Remark. 1 The cone of sums-of-squares of elements of V is a projection of a spectrahedron and, 2 solving linear optimization problems over affine slices of these cone reduces to semidefinite programming. The best of all worlds Every sum-of-squares is nonnegative. The purpose of these two lectures is to describe when do nonnegative polynomials coincide with sums-of-squares and to give a geometric explanation of this miracle. Example 1 Definition. a1 an The degree of the monomial X1 ::: Xn is a1 + a2 + ··· + an.A polynomial is homogeneous of degree d (i.e. a form of degree d) if it is a sum of monomials of the same degree d. Example: 2 4 f (X0; X1; X2; X3) = X0 X1X2 + X0X1X2X3 + X3 is a homogeneous polynomial of degree four in the variables X0; X1; X2; X3. Question. For which degrees 2d and number of variables n is every nonnegative homogeneous polynomial of degree 2d a sum-of-squares? Question. For which degrees 2d and number of variables n is every nonnegative homogeneous polynomial of degree 2d a sum-of-squares? Remark. 2 2 If F = s1 + ··· + sk is a homogeneous polynomial of degree k which is a sum-of-squares then: 1 The degree is even k = 2d. 2 The si are homogeneous polynomials of degree d. Example 1 Theorem. (Hilbert 1888) Every nonnegative form (i.e. homogeneous polynomial) of degree 2d in in n-variables is a sum-of-squares of forms of degree d if and only if either, 1 n = 2 (bivariate forms) or 2 d = 1 (quadratic forms) or 3 n = 3 and d = 2 (ternary quartics). Question. Is the following polynomial nonnegative? 4 2 2 4 2 2 2 2 2 4 f = 16x0 − 31x0 x1 + 25x1 − 30x0x1 x2 − 16x0 x2 + 45x1 x2 + 4x2 Example 1 Question. Is the following polynomial nonnegative? 4 2 2 4 2 2 2 2 2 4 f = 16x0 − 31x0 x1 + 25x1 − 30x0x1 x2 − 16x0 x2 + 45x1 x2 + 4x2 Yes. We can obtain the following sum-of-squares certificate of nonnegativity, 2 2 2 2 2 f = (3x0x1 − 5x1x2) + (4x0 − 5x1 − 2x2 ) Example 2: Multihomogeneous polynomials Definition. Let x0;:::; xn1 and y0;:::; yn2 be two sets of variables. The a1 an1 b1 bn2 bi-degree of the monomial x1 ::: xn1 y1 ::: yn2 is (a1 + ··· + an1 ; b1 + ··· + bn2 ). A polynomial F (x0;:::; xn1 ; y0 :::; yn2 ) is bi-homogeneous of degree (d1; d2) if all its monomials have degree vector (d1; d2). Example: 2 3 2 The monomial x0 x1 x2y0 y1y2y3 has degree vector (6; 5) in x0; x1; x2 and y0; y1; y2; y3. Example: The following polynomial is bi-homogeneous of degree (4; 2) in x0; x1 and y0; y1; y2. 2 2 2 3 2 4 2 3 2 2 2 g = x0 x1 y0 − 2x0x1 y0 + x1 y0 + 2x0 x1y0y1 + 4x0 x1 y2 Example 2: Multihomogeneous polynomials Question. For which bi-degrees (2d1; 2d2) and numbers of variables (n1; n2) do nonnegative bi-homogeneous polynomials and sums-of-squares (of bi-forms (d1; d2)) coincide? Example 2: Multihomogeneous polynomials Theorem. (Choi-Lam-Reznick 1980) A bi-homogeneous nonnegative polynomial of degree (2d1; 2d2) in two sets of variables of sizes n1; n2 ≥ 2 is a sum-of-squares of bi-homogeneous forms if and only if either (n1 = 2; d2 = 1) or (d1 = 1; n2 = 2) . Example 2: Multihomogeneous polynomials Theorem. (Choi-Lam-Reznick 1980) A bi-homogeneous nonnegative polynomial of degree (2d1; 2d2) in two sets of variables of sizes n1; n2 ≥ 2 is a sum-of-squares of bi-homogeneous forms if and only if either (n1 = 2; d2 = 1) or (d1 = 1; n2 = 2). Remark. Coincidence occurs only for forms which are binary in one set of variables and quadratic in the other. Example 3: Quadrics Theorem. (S-Lemma of Yakubovich 1971) Let Q(x1;:::; xn) be an indefinite quadratic form and let Z be the n set of zeroes of Q in R . Every quadratic form F which is nonnegative on Z is of the form 2 2 F = `1 + ··· + `k + λQ for some linear forms `i (x1;:::; xn) and some real number λ. Example 3: Quadrics Theorem. (S-Lemma of Yakubovich 1971) Let Q(x1;:::; xn) be an indefinite quadratic form and let Z be the n set of zeroes of Q in R . Every quadratic form F which is nonnegative on Z is of the form 2 2 F = `1 + ··· + `k + λQ for some linear forms `i (x1;:::; xn) and some real number λ. Remark. On Z every nonnegative polynomial F satisfies 2 2 F = `1 + ··· + `k Wish list... We would like to construct a theory of homogeneous sums-of-squares generalizing the previous equalities. It should cover two cases: 1 Homogeneous polynomials with prescribed support sets 2∆. Question. For which sets of monomials ∆ does it happen that every nonnegative polynomial supported in 2∆ is a sum-of-squares of elements of ∆? n 2 Polynomials restricted to special subsets Z ⊆ R Question. n For which Z ⊆ R does it happen that every homogeneous polynomial of degree 2d which is nonnegative on Z is equal to a sum-of-squares on Z? Looking for the best worlds For such a theory we need an appropriately general context to speak about homogeneous nonnegativity and sums-of-squares. Our current belief is that this context is that of real projective varieties. Worlds To a homogeneous system of equations F1 = 0;:::; Fm = 0 with Fi 2 R[X0;:::; Xn] we will associate two objects: n 1 (A geometric object.) A set X ⊆ P called a projective variety. 2 (An algebraic object.) A graded ring R[X ] called a homogeneous coordinate ring. Which will allow us to speak about nonnegative polynomials / sums-of-squares on X . Projective space Let n be a positive integer. Definition. n Projective n-space, denoted P is the set of lines through the origin n+1 in C . More precisely: n+1 1 On C n f0g define the equivalence relation u ∼ v iff n n+1 9λ 2 C (u = λv) and let P := C n f0g= ∼. n 2 The points of P are denoted by their homogeneous coordinates [α0 : :::: : αn] := [(α0; : : : ; αn)] A point [~α] has many affine representatives ~α Example: 2 In P we have [1 : 2 : 3] = [1=2 : 1 : 3=2] Projective varieties If F1;:::; Fm are homogeneous polynomials of positive degree let n V (F1;:::; Fm) := f[α] 2 P : Fi (α) = 0; i = 1;:::; mg Definition. n A projective variety is any set X ⊆ P of the form n X = V (F1;:::; Fm) (i.e. the solution set in P of a system of homogeneous polynomials). Real varieties If X = V (F1;:::; Fm) for Fi 2 R[X0;:::; Xn] homogeneous (i.e. if it is defined by equations with real coefficients) then we say that X is a real variety. Real coordinate rings Assume X = V (F1;:::; Fm) is a real variety. Definition. The real homogeneous coordinate ring of X is R[X ] := R[X0;:::; Xn]=I (X ) where I (X ) = fg 2 R[X0;:::; Xn]: g(α) = 0 for all [α] 2 X g Real coordinate rings Definition. The real homogeneous coordinate ring of X is R[X ] := R[X0;:::; Xn]=I (X ) with I (X ) = fg 2 R[X0;:::; Xn]: g(α) = 0 for all [α] 2 X g Remark. 1 In R[X ] we have p = q () p − q 2 I (X ). Therefore we can think of R[X ] as the ring of polynomials restricted to X . Example Question. 2 2 2 2 If F (A; B; C) = B − A − C then what is the variety X ⊆ P defined by F and what is the homogeneous coordinate ring of X ? X = V (B2 − A2 − C 2) ⊆ P2 A B X \fC 6= 0g = f[x : y : 1] : y 2−x2−1 = 0g with x := and y := C C X = V (B2 − A2 − C 2) The homogeneous coordinate ring of X is 2 2 2 R[X ] := R[A; B; C]=(B − A − C ) In R[X ] the following equality holds 2B2 − C 2 = B2 + (B2 − C 2) = B2 + A2 Nonnegative forms and Sums-of-squares on varieties n Let X ⊆ P be a real projective variety with homogeneous coordinate ring R[X ].
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