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

• f 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 pn(x, y) = xn + yn α = x + y β = xy. x, y are the roots of the equation λ2 − αλ + β = 0.

The first few Girard-Vi` ete formulae

p1 = α p2 = α2 − 2β p3 = α3 − 3αβ p4 = α4 − 4α2β + 2β2. Further formulae are obtained from the recursion pn+2 = αpn+1 − βpn.

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. 1798: Trait´ e de la R´ esolution des ´ Equations Num´ eriques de tous les Degr´ es, by Joseph Louis Lagrange.

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 pn = xn+yn, for integer n, can be written as free polynomials in some finite collection of ‘elementary symmetric functions’

• f x and y.

Nevertheless, we do find three free polynomials α, β, γ in x and y such that every pn can be written as a free polynomial in α, β, γ and β−1.

A free Newton-Girard-Vieta formula

Let u = 1

2(x + y),

v = 1

2(x − y)

and let α = u, β = v2, 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 Pn in three variables such that pn(x, y) = Pn(α, β, γ). Moreover Pn can be written as a free polynomial in α, β, γ and β−1.

Proof

Let qn = xn − yn. For any integer n, pn = xxn−1 + yyn−1 = (u + v)xn−1 + (u − v)yn−1 = u(xn−1 + yn−1) + v(xn−1 − yn−1) = upn−1 + vqn−1. Similarly, qn = vpn−1 + uqn−1. Thus

• pn

qn

• = T
• pn−1

qn−1

• where

T =

• u

v v u

• .

Hence

• pn

qn

• = T n
• p0

q0

• = T n
• 2
• .

Proof – continued

Define the free polynomials sn

even, sn

• dd 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. sn

even and sn

• dd are symmetric and antisymmetric respectively

as polynomials in x, y. By induction, for n ≥ 1, T n =

• sn

even

sn

• dd

sn

• dd

sn

even

• .

Hence pn = 2sn

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 v2 by β. If there are k u’s between the first and second v’s, replace vukv by (γβ−1)k−1γ. Continue until all u’s and v’s have been replaced. Hence pn = 2sn

even is a sum of monomials in α, β, γ and β−1.

The first few Pn

xn + yn = Pn(α, β, γ) where α = 1

2(x+y), β = 1 4(x−y)2, γ = 1 8(x−y)(x+y)(x−y).

P1 = 2α P2 = 2(α2 + β) P3 = 2(α3 + αβ + γ + βα) P4 = 2(α4 + α2β + αγ + γβ−1γ + αβα + γα + βα2 + β2) P−1 = 2(α − βγ−1β)−1 P−2 = 2

• α2 + β − (αβ + γ)(γβ−1γ + β2)−1(βα + γ)

−1

P−3 = 2

• α3 + αβ + βα + γ − (α2β + αγ + γβ−1γ + β2)×

(γβ−1γβ−1γ + γβ + βγ + βαβ)−1(βα2 + γα + γβ−1γ + β2)

−1 .

A free Waring-Lagrange theorem

Let u = 1

2(x + y),

v = 1

2(x − y)

and let α = u, β = v2, 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 Symd be the space of symmetric homogeneous polynomials of degree d. Then dim Symd = 2d−1. Let Qd ⊆ Symd comprise the polynomials in u, v2, vuv, . . . , vud−2v that are homogeneous of degree d in u, v, and hence also in x, y. Then Q1 = Cu and Qd ⊆ Symd. By induction dim Qd = 2d−1 = dim Symd, whence Qd = Symd.

Proof – conclusion

Observe that vu2v = vuv(v2)−1vuv = γβ−1γ vu3v = vuv(v2)−1vuv(v2)−1vuv = γβ−1γβ−1γ 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-

• rem 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(3zwz − izzw) 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 f :

∞

• n=1

M2

n → ∞

• n=1

Mn where Mn denotes the algebra of n × n matrices over C.

The nc universe

The nc analogue of Cd is Md def =

∞

• n=1

(Mn)d. ⊕ defines a binary operation on Md: if x ∈ Mn and y ∈ Mm then x ⊕ y def =

• x

y

• ∈ Mn+m.

If x = (x1, . . . , xd) and y = (y1, . . . , yd) are in Md then x ⊕ y def = (x1 ⊕ y1, . . . , xd ⊕ yd) ∈ Md. Similarities: if s ∈ GLn(C) and x ∈ Md

n then

s−1xs def = (s−1x1s, . . . , s−1xds) ∈ Md

n.

Properties of the function f(x1, x2) = exp(3x1x2x1 − ix1x1x2)

The function f : M2 → M1 has three important properties. (1) f is graded: if x ∈ M2

n then f(x) ∈ Mn.

(2) f preserves direct sums: f(x ⊕ y) = f(x) ⊕ f(y) for all x, y ∈ M2. (3) f preserves similarities: if s ∈ GLn(C) and x ∈ M2

n then

f(s−1xs) = s−1f(x)s.

nc functions

An nc set is a subset of Md that is closed under ⊕. An nc function is a function f defined on an nc set D ⊂ Md which is graded and preserves direct sums and similarities. Thus, if x ∈ D ∩ Md

n, s ∈ GLn(C) and s−1xs ∈ D then

f(s−1xs) = s−1f(x)s. Every free polynomial (that is, polynomial over C in d non- commuting indeterminates) defines an nc function on Md. An nc function f on D is analytic if D is open in the disjoint union topology on Md and f|D ∩ Md

n is analytic for every n.

Try to extend classical function theory to nc functions.

The free topology on Md

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

Free holomorphy

A function f on a set D ⊂ Md is freely holomorphic if (1) D is a freely open set in Md (2) f is a freely locally nc function D → M1 (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 Uα of X to a set Dα ⊂ Md. For charts α, β the transition map Tαβ : α(Uα ∩Uβ) → β(Uα ∩ Uβ) is Tαβ = β ◦ α−1. A is a d-dimensional nc atlas for X if {Uα : α ∈ A} covers X and, for all α, β ∈ A, (1) α(Uα ∩ Uβ) is a union of nc sets and (2) the restriction of Tαβ to any nc subset of α(Uα ∩ Uβ) 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 Md and the transition maps Tαβ are freely holomorphic for every α, β ∈ A. A map f : X → M1 is a freely holomorphic function on the free manifold (X, T , A) if f ◦ α−1 is a freely holomorphic function on Dα for every α ∈ A.

Waring and Lagrange again

In the commutative case, let π : C2 → C2 be given by π(z, w) = (z + w, zw). If f : C2 → C is a symmetric analytic function, then there exists an analytic function F on C2 such that f = F ◦ π.

An analytic free Waring-Lagrange theorem

There exists a two-dimensional Zariski-free manifold G and a holomorphic map π : M2 → G with the following property. There is a canonical bijection between the classes of (i) freely holomorphic symmetric nc functions f on M2, and (ii) holomorphic functions F defined on the manifold G that are conditionally nc and are such that, for every w ∈ M2, there is a free neighborhood U of w such that F is bounded

• n π(U) ∩ G.

Reference

Jim Agler, John E. McCarthy and N. J. Young, Non-commutative manifolds, the free square root and sym- metric functions in two non-commuting variables,

• Trans. London Math. Soc. (2018) 5(1) 132 – 183

The end

www1.maths.leeds.ac.uk/∼nicholas/slides/2019/NewtonGirard.pdf