The multiplicative sequence of a power series #
Let Q(t) = ∑ qₖ tᵏ be a power series over a commutative ring R. For n variables
x₁, …, xₙ, the coefficient of tᵏ in ∏ᵢ Q(xᵢ t) is a symmetric polynomial in the xᵢ, so
by the fundamental theorem of symmetric polynomials it is a unique polynomial
Kₖ(e₁, …, eₙ) in the elementary symmetric polynomials. When Q(0) = 1, the family (Kₖ) is
Hirzebruch's multiplicative sequence belonging to Q.
In characteristic class theory the xᵢ are the formal roots of a vector bundle and the eᵢ its
Chern classes: for Q(t) = t / (1 - e⁻ᵗ) the value Kₖ(c₁, …, cₙ) is the degree-2k part of
the Todd class of a complex bundle of rank n. The L and  classes come from even series
Q(z): one first writes Q(z) = P(z²) and then applies the construction to P, whose roots are
the squares xᵢ², so that the eᵢ are the elementary symmetric polynomials in the xᵢ², that
is, the Pontryagin classes. This file develops the algebra that such definitions rest on: the
root description of Kₖ in any commutative R-algebra, the Whitney (product) formula for roots,
the normalization on a single root, stability under adjoining a zero root, the root description
for even series, and the explicit polynomials K₁, K₂ and K₃.
Main definitions #
PowerSeries.multiplicativeSequence Q n k: the polynomialKₖinnvariables, where the variablej : Fin nstands for the elementary symmetric polynomiale_{j+1}.
Main results #
PowerSeries.aeval_esymm_multiplicativeSequenceandPowerSeries.eq_multiplicativeSequence_iff:Kₖis the unique polynomial whose value ate₁, …, eₙis the coefficient oftᵏin∏ᵢ Q(xᵢ t).PowerSeries.mk_aeval_multiplicativeSequence: fornrootsxᵢin any commutativeR-algebra,∑ₖ Kₖ(e₁(x), …, eₙ(x)) tᵏ = ∏ᵢ Q(xᵢ t).PowerSeries.aeval_multiset_esymm_multiplicativeSequence: the same, coefficientwise.PowerSeries.aeval_multiset_esymm_add_multiplicativeSequence: the Whitney formula: on the union of two families of roots,Kₖis the convolution of the values ofKon each family.PowerSeries.multiplicativeSequence_one_left: on a single root,Kₖ(e₁) = qₖ e₁ᵏ.PowerSeries.multiplicativeSequence_zero_right:K₀ = Q(0) ^ n.PowerSeries.aeval_snoc_zero_multiplicativeSequence: settingeₙ₊₁ = 0in the polynomial forn + 1roots givesQ(0)times the polynomial fornroots.PowerSeries.aeval_pi_single_zero_multiplicativeSequence: evaluating ate₁ = xandeⱼ = 0forj ≥ 2givesQ(0) ^ n qₖ xᵏ; whenQ(0) = 1, the total class1 + xgoes toQ(x).PowerSeries.expand_mk_aeval_multiplicativeSequence,PowerSeries.aeval_multiset_esymm_sq_multiplicativeSequence: for the even seriesQ(t²),∏_{x ∈ s} Q(x² t²)is∑ₖ Kₖ(e₁(s²), …, eₙ(s²)) t²ᵏ, wheres²is the multiset of squares.PowerSeries.multiplicativeSequence_one_right,PowerSeries.multiplicativeSequence_two_right,PowerSeries.multiplicativeSequence_three_right: forQ(0) = 1,K₁ = q₁ e₁,K₂ = q₂ e₁² + (q₁² - 2 q₂) e₂andK₃ = q₃ e₁³ + (q₁ q₂ - 3 q₃) e₁ e₂ + (q₁³ - 3 q₁ q₂ + 3 q₃) e₃.
References #
- F. Hirzebruch, Topological Methods in Algebraic Geometry, §1.
- J. Milnor and J. Stasheff, Characteristic Classes, §19.
Pushing ∏ᵢ Q(xᵢ t) forward along an algebra homomorphism φ out of R[x] replaces each
root xᵢ by φ xᵢ.
The coefficient of tᵏ in ∏ᵢ Q(xᵢ t) is a symmetric polynomial in the xᵢ.
The multiplicative sequence belonging to the power series Q, in n variables:
Q.multiplicativeSequence n k is the unique polynomial Kₖ such that substituting the
elementary symmetric polynomials e₁, …, eₙ of x₁, …, xₙ gives the coefficient of tᵏ in
∏ᵢ Q(xᵢ t). The variable j : Fin n stands for e_{j+1}, following
MvPolynomial.esymmAlgHom.
Instances For
The defining property of PowerSeries.multiplicativeSequence: its value at the elementary
symmetric polynomials e₁, …, eₙ is the coefficient of tᵏ in ∏ᵢ Q(xᵢ t).
PowerSeries.multiplicativeSequence is characterized by its value at the elementary
symmetric polynomials.
Root description of the multiplicative sequence: for a multiset s of n roots in a
commutative R-algebra, ∑ₖ Kₖ(e₁(s), …, eₙ(s)) tᵏ = ∏_{x ∈ s} Q(x t).
Root description of the multiplicative sequence, coefficientwise: for a multiset s of
n roots, Kₖ(e₁(s), …, eₙ(s)) is the coefficient of tᵏ in ∏_{x ∈ s} Q(x t).
The Whitney formula for roots: on the union of a family s of n roots and a family t
of m roots, Kₖ is the convolution of the values of the multiplicative sequence on s and
on t.
In a single variable, e₁ is the root itself, so Kₖ(e₁) = qₖ e₁ᵏ.
The degree-zero term of the multiplicative sequence is Q(0) ^ n; in particular it is 1
when Q(0) = 1.
Stability of the multiplicative sequence: setting eₙ₊₁ = 0 in the polynomial for
n + 1 roots gives Q(0) times the polynomial for n roots. When Q(0) = 1, the
multiplicative sequence is therefore unchanged by adjoining a zero root.
Normalization of the multiplicative sequence: evaluating at e₁ = x and eⱼ = 0 for
j ≥ 2 gives Q(0) ^ n qₖ xᵏ. When Q(0) = 1, the multiplicative sequence therefore sends the
total class 1 + x to Q(x).
Multiplicative sequences of even series: for a multiset s of n roots,
∏_{x ∈ s} Q(x² t²) is obtained from ∑ₖ Kₖ(e₁(s²), …, eₙ(s²)) tᵏ, where Kₖ is the
multiplicative sequence of Q and s² is the multiset of squares of the roots, by substituting
t² for t. An even characteristic series is of the form Q(t²); for the L and  classes the
roots are one degree-two root per oriented real 2-plane, and the eᵢ(s²) are the Pontryagin
classes.
Multiplicative sequences of even series, coefficientwise: for a multiset s of n roots,
Kₖ(e₁(s²), …, eₙ(s²)) is the coefficient of t²ᵏ in ∏_{x ∈ s} Q(x² t²), where Kₖ is the
multiplicative sequence of Q.
The linear term of a multiplicative sequence: for Q(0) = 1, K₁ = q₁ e₁.
The quadratic term of a multiplicative sequence: for Q(0) = 1 and at least two variables,
K₂ = q₂ e₁² + (q₁² - 2 q₂) e₂.
The cubic term of a multiplicative sequence: for Q(0) = 1 and at least three variables,
K₃ = q₃ e₁³ + (q₁ q₂ - 3 q₃) e₁ e₂ + (q₁³ - 3 q₁ q₂ + 3 q₃) e₃.