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 ∈ R[X1,..., Xn] be a multivariate polynomial with real coeﬃcients. Deﬁnition. n The polynomial f is nonnegative if f (α) ≥ 0 for every α ∈ R

Deﬁnition. The polynomial f is a sum-of-squares if there exist an integer t > 0 and polynomials g1,..., gt ∈ 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) α∈Rn Nonnegative polynomials

Deﬁnition. The global polynomial optimization problem asks us to ﬁnd

α∗ := inf f (α) α∈Rn

Remark.

∗ α = sup{λ ∈ R : f (x) − λ ≥ 0} Nonnegative polynomials

Deﬁnition. The global polynomial optimization problem asks us to ﬁnd

α∗ := inf f (α) α∈Rn

Remark.

∗ α = sup{λ ∈ R : f (x) − λ ≥ 0} 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 ∈ 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

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)

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 ∈ 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 \{0} define the equivalence relation u ∼ v iff n n+1 ∃λ ∈ C (u = λv) and let P := C \{0}/ ∼. 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) := {[α] ∈ P : Fi (α) = 0, i = 1,..., m}

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 ∈ 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 ) = {g ∈ R[X0,..., Xn]: g(α) = 0 for all [α] ∈ X } Real coordinate rings

Deﬁnition. The real homogeneous coordinate ring of X is

R[X ] := R[X0,..., Xn]/I (X )

with I (X ) = {g ∈ R[X0,..., Xn]: g(α) = 0 for all [α] ∈ X }

Remark. 1 In R[X ] we have p = q ⇐⇒ p − q ∈ 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 ∩{C 6= 0} = {[x : y : 1] : y 2−x2−1 = 0} 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 ]. Let R[X ]2 be the quadratic forms restricted to X . Deﬁnition. The cone of nonnegative quadratic forms on X

PX = {g ∈ R[X ]2 : ∀[α] ∈ X (R)(g(α) ≥ 0)}

Deﬁnition. The cone of sums-of-squares of linear forms on X

2 2 ΣX := g ∈ R[X ]2 : ∃t ∈ N, `1, . . . , `t ∈ R[X ]1 g = `1 + ··· + `t Main question

It is immediate that ΣX ⊆ PX . As before PX allows us to model many problems of interests and ΣX is a projection of a spectrahedron. We ask: Question. n For which real varieties X ⊆ P does the equality PX = ΣX hold? In principle, restricting only to quadratic forms seems to be a very strong limitation. However, as we will see, this is not the case since we are considering arbitrary varieties, thanks to a classical linearization trick, the Veronese re-embedding of varieties. The Veronese embeddings of Pn

Given integers d and n let m0,..., mk be an enumeration of all monomials of degree d in X0,..., Xn. Deﬁnition. n k The d-uple map νd : P → P is given by the formula

νd ([α0 : ··· : αn]) = [m0 : ··· : mk ].

n n k The d-th Veronese embedding of P is X := νd (P ) ⊆ P with n+d k = d − 1. The Veronese embeddings of Pn

Example: 2 5 Set n = d = 2. The Veronese surface X := ν2(P ) ⊆ P is the 2 image of P under the map

2 2 2 5 ν2([A : B : C]) = [A : AB : BC : B : BC : C ] ⊆ P Properties of the Veronese maps νd

There are two facts you should keep in mind: n k n 1 νd : P → X ⊆ P is an isomorphism where X := νd (P ). 2 Composing with the d-th Veronese map transforms a form of k n degree j ∈ P into one of degree dj in P .

n Linearization trick. If Y ⊆ P and we want to study forms of degree 2d in Y then we can study quadratic forms in k X := νd (Y ) ⊆ P instead. Question. n For which real varieties X ⊆ P does the equality PX = ΣX hold? Simplifying Assumptions (WLOG)

n A hyperplane H ⊆ P is H = V (a0X0 + ··· + anXn). Deﬁnition. n A variety X ⊆ P is non-degenerate if it is not contained in any n hyperplane H ⊆ P . Simplifying Assumptions (WLOG)

n If X ⊆ P is a variety we denote by X (R) ⊆ X the real points of X (i.e. points with real homogeneous coordinates). Deﬁnition. n A variety X ⊆ P is totally real if every homogeneous polynomial vanishing in X (R) vanishes on X . Simplifying Assumptions (WLOG)

Remark. In that case X is the smallest variety containing X (R) (equiv. X (R) is Zariski-dense in X ). Main question

Question. Assume X is non-degenerate and totally real. When does the equality PX = ΣX hold? Main Theorem – a partial answer.

n Let X ⊆ P be a real projective variety. Assume: 1 X is non-degenerate and totally real. 2 X is irreducible (i.e. is not the union of ﬁnitely-many subvarieties)

Theorem. (Blekherman, Smith, - , 2016)

The equality PX = ΣX occurs if and only if the following equality holds: deg(X ) = 1 + codim(X) Main Theorem – a partial answer.

n Let X ⊆ P be a real projective variety. Assume: 1 X is non-degenerate and totally real. 2 X is irreducible (i.e. is not the union of ﬁnitely-many subvarieties)

Theorem. (Blekherman, Smith, - , 2016)

The equality PX = ΣX occurs if and only if the following equality holds: deg(X ) = 1 + codim(X)

The dimension dim(X ) (equiv. codim(X) := n − dim(X)) and the degree deg(X ) are the main numerical invariants of a projective variety X .