Low-rank sums-of-squares representations Cynthia Vinzant, North - PowerPoint PPT Presentation

Low-rank sums-of-squares representations Cynthia Vinzant, North Carolina State University joint work with Greg Blekherman, Daniel Plaumann, and Rainer Sinn JMM 2017 Cynthia Vinzant Low-rank sums-of-squares representations

Sums of squares and nonnegative polynomials A representation of a element f ∈ R as a sum of squares over a ring R (usually R [ x 0 , . . . , x n ] or a quotient) is an expression f = h 2 1 + . . . + h 2 where h j ∈ R . r Cynthia Vinzant Low-rank sums-of-squares representations

Sums of squares and nonnegative polynomials A representation of a element f ∈ R as a sum of squares over a ring R (usually R [ x 0 , . . . , x n ] or a quotient) is an expression f = h 2 1 + . . . + h 2 where h j ∈ R . r Over R = R [ x 0 , . . . , x n ], this certifies the nonnegativity of f on R n +1 . e.g. x 4 − 4 x 3 y + 5 x 2 y 2 − 2 xy 3 + y 4 Cynthia Vinzant Low-rank sums-of-squares representations

Sums of squares and nonnegative polynomials A representation of a element f ∈ R as a sum of squares over a ring R (usually R [ x 0 , . . . , x n ] or a quotient) is an expression f = h 2 1 + . . . + h 2 where h j ∈ R . r Over R = R [ x 0 , . . . , x n ], this certifies the nonnegativity of f on R n +1 . e.g. x 4 − 4 x 3 y + 5 x 2 y 2 − 2 xy 3 + y 4 = ( x 2 − 2 xy ) 2 + ( xy − y 2 ) 2 Cynthia Vinzant Low-rank sums-of-squares representations

Sums of squares and nonnegative polynomials A representation of a element f ∈ R as a sum of squares over a ring R (usually R [ x 0 , . . . , x n ] or a quotient) is an expression f = h 2 1 + . . . + h 2 where h j ∈ R . r Over R = R [ x 0 , . . . , x n ], this certifies the nonnegativity of f on R n +1 . e.g. x 4 − 4 x 3 y + 5 x 2 y 2 − 2 xy 3 + y 4 = ( x 2 − 2 xy ) 2 + ( xy − y 2 ) 2 Let Σ n , 2 d denote the sums of squares in R [ x 0 , . . . , x n ] 2 d and P n , 2 d denote polynomials in R [ x 0 , . . . , x n ] 2 d nonnegative on R n +1 . Cynthia Vinzant Low-rank sums-of-squares representations

Sums of squares and nonnegative polynomials A representation of a element f ∈ R as a sum of squares over a ring R (usually R [ x 0 , . . . , x n ] or a quotient) is an expression f = h 2 1 + . . . + h 2 where h j ∈ R . r Over R = R [ x 0 , . . . , x n ], this certifies the nonnegativity of f on R n +1 . e.g. x 4 − 4 x 3 y + 5 x 2 y 2 − 2 xy 3 + y 4 = ( x 2 − 2 xy ) 2 + ( xy − y 2 ) 2 Let Σ n , 2 d denote the sums of squares in R [ x 0 , . . . , x n ] 2 d and P n , 2 d denote polynomials in R [ x 0 , . . . , x n ] 2 d nonnegative on R n +1 . Theorem (Hilbert): Σ n , 2 d = P n , 2 d if and only if n = 1 or 2 d = 2 or ( n , 2 d ) = (2 , 4) . Motzkin non-example: x 2 y 4 + x 4 y 2 − 3 x 2 y 2 z 2 + z 6 ∈ P 2 , 6 \ Σ 2 , 6 Cynthia Vinzant Low-rank sums-of-squares representations

Number of squares n = 1 : A nonnegative bivariate form is a sum of two squares Proof: Factor f = ( p + ✐ q )( p − ✐ q ) = p 2 + q 2 where p , q ∈ R [ x 0 , x 1 ] d 2 d = 2 : A nonnegative quadratic form in P n , 2 is a sum of n + 1 squares Proof: Diagonalization of quadratic forms ( n , 2 d ) = (2 , 4) : A nonnegative ternary quartic is a sum of three squares Proof by Hilbert, 1888 Cynthia Vinzant Low-rank sums-of-squares representations

Number of squares n = 1 : A nonnegative bivariate form is a sum of two squares Proof: Factor f = ( p + ✐ q )( p − ✐ q ) = p 2 + q 2 where p , q ∈ R [ x 0 , x 1 ] d 2 d = 2 : A nonnegative quadratic form in P n , 2 is a sum of n + 1 squares Proof: Diagonalization of quadratic forms ( n , 2 d ) = (2 , 4) : A nonnegative ternary quartic is a sum of three squares Proof by Hilbert, 1888 Our goal: Unify/generalize these results using varieties of minimal degree Cynthia Vinzant Low-rank sums-of-squares representations

Quadratic forms on varieties Let . . . Zar , X ⊂ P N ( C ) = a real, nondegenerate irreducible variety equal to X ( R ) R [ X ] k = R [ x 0 , . . . , x N ] k / I ( X ) = coordinate ring of X in degree k , Cynthia Vinzant Low-rank sums-of-squares representations

Quadratic forms on varieties Let . . . Zar , X ⊂ P N ( C ) = a real, nondegenerate irreducible variety equal to X ( R ) R [ X ] k = R [ x 0 , . . . , x N ] k / I ( X ) = coordinate ring of X in degree k , Σ X = { ℓ 2 1 + . . . + ℓ 2 r : ℓ i ∈ R [ X ] 1 } ⊂ R [ X ] 2 , and P X = { q ∈ R [ X ] 2 : q ( x ) ≥ 0 for all x ∈ X ( R ) } ⊂ R [ X ] 2 Cynthia Vinzant Low-rank sums-of-squares representations

Quadratic forms on varieties Let . . . Zar , X ⊂ P N ( C ) = a real, nondegenerate irreducible variety equal to X ( R ) R [ X ] k = R [ x 0 , . . . , x N ] k / I ( X ) = coordinate ring of X in degree k , Σ X = { ℓ 2 1 + . . . + ℓ 2 r : ℓ i ∈ R [ X ] 1 } ⊂ R [ X ] 2 , and P X = { q ∈ R [ X ] 2 : q ( x ) ≥ 0 for all x ∈ X ( R ) } ⊂ R [ X ] 2 If X = ν d ( P n ), then Σ X ∼ = Σ n , 2 d and P X ∼ = P n , 2 d . Cynthia Vinzant Low-rank sums-of-squares representations

Quadratic forms on varieties Let . . . Zar , X ⊂ P N ( C ) = a real, nondegenerate irreducible variety equal to X ( R ) R [ X ] k = R [ x 0 , . . . , x N ] k / I ( X ) = coordinate ring of X in degree k , Σ X = { ℓ 2 1 + . . . + ℓ 2 r : ℓ i ∈ R [ X ] 1 } ⊂ R [ X ] 2 , and P X = { q ∈ R [ X ] 2 : q ( x ) ≥ 0 for all x ∈ X ( R ) } ⊂ R [ X ] 2 If X = ν d ( P n ), then Σ X ∼ = Σ n , 2 d and P X ∼ = P n , 2 d . Ex: Let X = ν 2 ( P 1 ), where ν 2 ([ x : y ]) = [ x 2 : xy : y 2 ] = [ a : b : c ]. Then ( x 2 − 2 xy ) 2 + ( xy − y 2 ) 2 ∈ Σ 1 , 4 ↔ ( a − 2 b ) 2 + ( b − c ) 2 ∈ Σ X Cynthia Vinzant Low-rank sums-of-squares representations

Quadratic forms on varieties Let . . . Zar , X ⊂ P N ( C ) = a real, nondegenerate irreducible variety equal to X ( R ) R [ X ] k = R [ x 0 , . . . , x N ] k / I ( X ) = coordinate ring of X in degree k , Σ X = { ℓ 2 1 + . . . + ℓ 2 r : ℓ i ∈ R [ X ] 1 } ⊂ R [ X ] 2 , and P X = { q ∈ R [ X ] 2 : q ( x ) ≥ 0 for all x ∈ X ( R ) } ⊂ R [ X ] 2 If X = ν d ( P n ), then Σ X ∼ = Σ n , 2 d and P X ∼ = P n , 2 d . Ex: Let X = ν 2 ( P 1 ), where ν 2 ([ x : y ]) = [ x 2 : xy : y 2 ] = [ a : b : c ]. Then ( x 2 − 2 xy ) 2 + ( xy − y 2 ) 2 ∈ Σ 1 , 4 ↔ ( a − 2 b ) 2 + ( b − c ) 2 ∈ Σ X Theorem (Blekherman-Smith-Velasco): Σ X = P X if and only if X is a variety of minimal degree (i.e. deg( X ) = codim( X ) + 1). Cynthia Vinzant Low-rank sums-of-squares representations

Quadratic forms on varieties Let . . . Zar , X ⊂ P N ( C ) = a real, nondegenerate irreducible variety equal to X ( R ) R [ X ] k = R [ x 0 , . . . , x N ] k / I ( X ) = coordinate ring of X in degree k , Σ X = { ℓ 2 1 + . . . + ℓ 2 r : ℓ i ∈ R [ X ] 1 } ⊂ R [ X ] 2 , and P X = { q ∈ R [ X ] 2 : q ( x ) ≥ 0 for all x ∈ X ( R ) } ⊂ R [ X ] 2 If X = ν d ( P n ), then Σ X ∼ = Σ n , 2 d and P X ∼ = P n , 2 d . Ex: Let X = ν 2 ( P 1 ), where ν 2 ([ x : y ]) = [ x 2 : xy : y 2 ] = [ a : b : c ]. Then ( x 2 − 2 xy ) 2 + ( xy − y 2 ) 2 ∈ Σ 1 , 4 ↔ ( a − 2 b ) 2 + ( b − c ) 2 ∈ Σ X Theorem (Blekherman-Smith-Velasco): Σ X = P X if and only if X is a variety of minimal degree (i.e. deg( X ) = codim( X ) + 1). ν d ( P n ) has minimal degree ⇔ n = 1, d = 1, or ( n , d ) = (2 , 2) Corollary: Hilbert’s result. Cynthia Vinzant Low-rank sums-of-squares representations

Varieties of minimal degree Theorem: If X is a variety of minimal degree, then any q ∈ P X is a sum of dim( X ) + 1 squares. For generic q this bound is tight. Cynthia Vinzant Low-rank sums-of-squares representations

Varieties of minimal degree Theorem: If X is a variety of minimal degree, then any q ∈ P X is a sum of dim( X ) + 1 squares. For generic q this bound is tight. A variety of minimal degree is isomorphic to one of the following: ◮ a quadratic hypersurface ◮ ν d ( P 1 ) ◮ ν 2 ( P 2 ) ◮ a rational normal scroll ◮ a cone over one of the above. Cynthia Vinzant Low-rank sums-of-squares representations

Varieties of minimal degree Theorem: If X is a variety of minimal degree, then any q ∈ P X is a sum of dim( X ) + 1 squares. For generic q this bound is tight. A variety of minimal degree is isomorphic to one of the following: ◮ a quadratic hypersurface ◮ ν d ( P 1 ) ⇒ f ∈ P 1 , 2 d = sum of 2 squares ◮ ν 2 ( P 2 ) ◮ a rational normal scroll ◮ a cone over one of the above. Cynthia Vinzant Low-rank sums-of-squares representations

Varieties of minimal degree Theorem: If X is a variety of minimal degree, then any q ∈ P X is a sum of dim( X ) + 1 squares. For generic q this bound is tight. A variety of minimal degree is isomorphic to one of the following: ◮ a quadratic hypersurface ◮ ν d ( P 1 ) ⇒ f ∈ P 1 , 2 d = sum of 2 squares ◮ ν 2 ( P 2 ) ⇒ f ∈ P 2 , 4 = sum of 3 squares ◮ a rational normal scroll ◮ a cone over one of the above. Cynthia Vinzant Low-rank sums-of-squares representations

Varieties of minimal degree Theorem: If X is a variety of minimal degree, then any q ∈ P X is a sum of dim( X ) + 1 squares. For generic q this bound is tight. A variety of minimal degree is isomorphic to one of the following: ◮ a quadratic hypersurface ◮ ν d ( P 1 ) ⇒ f ∈ P 1 , 2 d = sum of 2 squares ◮ ν 2 ( P 2 ) ⇒ f ∈ P 2 , 4 = sum of 3 squares ◮ a rational normal scroll ⇒ ?? ◮ a cone over one of the above. Cynthia Vinzant Low-rank sums-of-squares representations