Newton-Girard-Vieta and Waring-Lagrange theorems for two - PowerPoint PPT Presentation

Newton-Girard-Vieta and Waring-Lagrange theorems for two non-commuting variables Nicholas Young Leeds and Newcastle Universities Joint work with Jim Agler, UCSD and John McCarthy, Washington University Lille, March 2019

Power sums In 1629 Albert Girard gave formulae for the sums of powers of the roots of a polynomial equation in terms of the coeffi- cients of the polynomial. In 1593 Fran¸ cois Vi` ete had given the case of polynomials with positive roots. The formulae were subsequently often attributed to Newton ( Algebra Universalis , 1707). Consider two commuting variables x, y . For any integer n let p n ( x, y ) = x n + y n α = x + y β = xy. x, y are the roots of the equation λ 2 − αλ + β = 0.

The first few Girard-Vi` ete formulae p 1 = α p 2 = α 2 − 2 β p 3 = α 3 − 3 αβ p 4 = α 4 − 4 α 2 β + 2 β 2 . Further formulae are obtained from the recursion p n +2 = αp n +1 − βp n .

Symmetric polynomials A polynomial p is symmetric if it is unchanged by a permu- tation of the variables. The Waring-Lagrange theorem Every symmetric polynomial is expressible as a polynomial in the elementary symmetric polynomials. Thus any symmetric polynomial in x and y can be written as a polynomial in x + y and xy . 1762: Meditationes Algebraicae , by Edward Waring, Lu- casian Professor at Cambridge. esolution des ´ 1798: Trait´ e de la R´ Equations Num´ eriques de es , by Joseph Louis Lagrange. tous les Degr´

What if x and y do not commute? A free polynomial is a polynomial in finitely many non- commuting variables. The symmetric free polynomial xyx + yxy cannot be written as a free polynomial in x + y and xy + yx . Show this by substituting 2 × 2 matrices for x, y in such a way that xy + yx = 0.

Theorem (Margarete Wolf, 1936) There is no finite basis for the algebra of free polynomials in d indeterminates over C when d > 1 . Thus there is no reason to expect that the free polynomials p n = x n + y n , for integer n , can be written as free polynomials in some finite collection of ‘elementary symmetric functions’ of x and y . Nevertheless, we do find three free polynomials α, β, γ in x and y such that every p n can be written as a free polynomial in α, β, γ and β − 1 .

A free Newton-Girard-Vieta formula Let u = 1 v = 1 2 ( x + y ) , 2 ( x − y ) and let β = v 2 , α = u, γ = vuv. Then α, β, γ are symmetric free polynomials in x, y , and, for every positive integer n , there exists a free rational function P n in three variables such that p n ( x, y ) = P n ( α, β, γ ) . Moreover P n can be written as a free polynomial in α, β, γ and β − 1 .

Proof Let q n = x n − y n . For any integer n , p n = xx n − 1 + yy n − 1 = ( u + v ) x n − 1 + ( u − v ) y n − 1 = u ( x n − 1 + y n − 1 ) + v ( x n − 1 − y n − 1 ) = up n − 1 + vq n − 1 . Similarly, q n = vp n − 1 + uq n − 1 . Thus � � � � � � p n p n − 1 u v = T where T = . q n q n − 1 v u Hence � � � � � � 2 p n p 0 = T n = T n . 0 q n q 0

Proof – continued Define the free polynomials s n even , s n odd in u, v for n ≥ 0 to be the sum of all monomials in u, v of total degree n and of even or odd degree respectively in v . s n even and s n odd are symmetric and antisymmetric respectively as polynomials in x, y . By induction, for n ≥ 1, � s n s n � T n = even odd . s n s n even odd Hence p n = 2 s n even .

Proof – conclusion Any monomial in u and v , in which v occurs with even de- gree, can be written as a monomial in α, β, γ and β − 1 . Starting at one end of the monomial, replace all the initial u ’s by α ’s. The first v must be followed by another (since the number of v ’s is even). If it is immediately following, replace v 2 by β . If there are k u ’s between the first and second v ’s, replace vu k v by ( γβ − 1 ) k − 1 γ . Continue until all u ’s and v ’s have been replaced. even is a sum of monomials in α, β, γ and β − 1 . Hence p n = 2 s n

The first few P n x n + y n = P n ( α, β, γ ) where α = 1 2 ( x + y ) , β = 1 4 ( x − y ) 2 , γ = 1 8 ( x − y )( x + y )( x − y ). P 1 = 2 α P 2 = 2( α 2 + β ) P 3 = 2( α 3 + αβ + γ + βα ) P 4 = 2( α 4 + α 2 β + αγ + γβ − 1 γ + αβα + γα + βα 2 + β 2 ) P − 1 = 2( α − βγ − 1 β ) − 1 � − 1 α 2 + β − ( αβ + γ )( γβ − 1 γ + β 2 ) − 1 ( βα + γ ) � P − 2 = 2 α 3 + αβ + βα + γ − ( α 2 β + αγ + γβ − 1 γ + β 2 ) × � P − 3 = 2 � − 1 . ( γβ − 1 γβ − 1 γ + γβ + βγ + βαβ ) − 1 ( βα 2 + γα + γβ − 1 γ + β 2 )

A free Waring-Lagrange theorem Let u = 1 v = 1 2 ( x + y ) , 2 ( x − y ) and let β = v 2 , α = u, γ = vuv. Every free polynomial in x and y can be written as a free polynomial in α, β, γ and β − 1 . It’s also true when ‘polynomial’ is replaced by ‘rational func- tion’.

Proof For d ≥ 1 let Sym d be the space of symmetric homogeneous polynomials of degree d . Then dim Sym d = 2 d − 1 . Let Q d ⊆ Sym d comprise the polynomials in u, v 2 , vuv, . . . , vu d − 2 v that are homogeneous of degree d in u, v , and hence also in x, y . Then Q 1 = C u and Q d ⊆ Sym d . By induction dim Q d = 2 d − 1 = dim Sym d , whence Q d = Sym d .

Proof – conclusion Observe that vu 2 v = vuv ( v 2 ) − 1 vuv = γβ − 1 γ vu 3 v = vuv ( v 2 ) − 1 vuv ( v 2 ) − 1 vuv = γβ − 1 γβ − 1 γ and so on. Hence every symmetric homogeneous free polynomial in x, y is expressible as a polynomial in α, β, γ and β − 1 .

Non-commutative analysis We wish to prove an analogue of the Waring-Lagrange the- orem for analytic functions of two non-commuting variables. We use the framework of non-commutative (or nc-) analy- sis introduced by J. L. Taylor in the 1970s and intensively developed over the last 10 years by many analysts.

What is an analytic function of noncommuting variables? The function f ( z, w ) = exp(3 zwz − i zzw ) looks like an analytic function of noncommuting variables z and w . How should we interpret this statement? J. L. Taylor, Functions of several non-commuting variables, Bull. AMS 79 (1973) interpreted f as a map ∞ ∞ M 2 � � f : n → M n n =1 n =1 where M n denotes the algebra of n × n matrices over C .

The nc universe The nc analogue of C d is ∞ M d def ( M n ) d . � = n =1 ⊕ defines a binary operation on M d : if x ∈ M n and y ∈ M m � � 0 x then x ⊕ y def = ∈ M n + m . 0 y If x = ( x 1 , . . . , x d ) and y = ( y 1 , . . . , y d ) are in M d then x ⊕ y def = ( x 1 ⊕ y 1 , . . . , x d ⊕ y d ) ∈ M d . Similarities: if s ∈ GL n ( C ) and x ∈ M d n then s − 1 xs def = ( s − 1 x 1 s, . . . , s − 1 x d s ) ∈ M d n .

Properties of the function f ( x 1 , x 2 ) = exp(3 x 1 x 2 x 1 − i x 1 x 1 x 2 ) The function f : M 2 → M 1 has three important properties. (1) f is graded : if x ∈ M 2 n then f ( x ) ∈ M n . (2) f preserves direct sums : f ( x ⊕ y ) = f ( x ) ⊕ f ( y ) for all x, y ∈ M 2 . (3) f preserves similarities : if s ∈ GL n ( C ) and x ∈ M 2 n then f ( s − 1 xs ) = s − 1 f ( x ) s.

nc functions An nc set is a subset of M d that is closed under ⊕ . An nc function is a function f defined on an nc set D ⊂ M d which is graded and preserves direct sums and similarities. n , s ∈ GL n ( C ) and s − 1 xs ∈ D then Thus, if x ∈ D ∩ M d f ( s − 1 xs ) = s − 1 f ( x ) s. Every free polynomial (that is, polynomial over C in d non- commuting indeterminates) defines an nc function on M d . An nc function f on D is analytic if D is open in the disjoint union topology on M d and f | D ∩ M d n is analytic for every n . Try to extend classical function theory to nc functions.

The free topology on M d For any I × J matrix δ = [ δ ij ] of free polynomials in d non- commuting variables define B δ = { x ∈ M d : � δ ( x ) � < 1 } . The free topology on M d is the topology for which a base consists of the sets B δ . The free topology is not Hausdorff. It does not distinguish between x and x ⊕ x . M d is connected in the free topology.

Free holomorphy A function f on a set D ⊂ M d is freely holomorphic if (1) D is a freely open set in M d (2) f is a freely locally nc function D → M 1 (3) f is freely locally bounded on D . Surprising theorem A freely holomorphic function is analytic.

nc manifolds Let X be a set. A d -dimensional nc chart on X is a bijective map α from a subset U α of X to a set D α ⊂ M d . For charts α, β the transition map T αβ : α ( U α ∩ U β ) → β ( U α ∩ U β ) is T αβ = β ◦ α − 1 . A is a d -dimensional nc atlas for X if { U α : α ∈ A} covers X and, for all α, β ∈ A , (1) α ( U α ∩ U β ) is a union of nc sets and (2) the restriction of T αβ to any nc subset of α ( U α ∩ U β ) is an nc map. ( X, A ) is a d -dimensional nc manifold if A is a d -dimensional nc atlas for X .

Free manifolds Let ( X, A ) be a d -dimensional nc manifold and let T be a topology on X . ( X, T , A ) is a d -dimensional free manifold if the range of every chart α ∈ A is freely open in M d and the transition maps T αβ are freely holomorphic for every α, β ∈ A . A map f : X → M 1 is a freely holomorphic function on the free manifold ( X, T , A ) if f ◦ α − 1 is a freely holomorphic function on D α for every α ∈ A .