When I define a polynomial, I am simply handing you an indexed collection of coefficients.

A polynomial with two variables, (x, y) and coefficients (c), is of the form:

(F(x, y) = \sum\limits_{ij} c_{ij} x^i y^j)

The coefficients of a polynomial form a ring. In other words, the coefficients (c_{ij}) are members of a coefficient ring (R). When we say (F) is over (R), we mean that (F) has coefficients in (R).

Example: The polynomial (F(x,y) = 7 + 5xy^2 + 2x^3) can be written out as (F(x,y) =7x^0y^0 + 5x^1y^2 + 2x^3y^0) such that (c_{00} = 7), (c_{12} = 5), (c_{30} = 2), and the rest of (c_{ij} = 0).

Alright, now let’s change the coefficients; reassign (c_{00} = 4), (c_{78} = 3), and all other (c_{ij} = 0).

Out pops a very different polynomial (P(x,y) = 4 + 3x^7y^8).

In other words, by altering the coefficients (c_{ij}) of (F(x,y)) via a ring homomorphism (u: R \to R’) (from the coefficient ring (c_{ij} \in R) to a coefficient ring (u(c_{ij}) \in R’))…

… we can get from (F(x,y)) to any other polynomial (F’(x,y)).

What’s a group-y polynomial?

Intuitively, a polynomial is “group-y” if there’s a constraint on our coefficients that forces the polynomial to satisfy the laws of a commutative group.

Concretely, a group-y polynomial is an operation of the form (F(x,y) = \sum\limits_{ij}c_{ij}x^iy^j) such that

1. commutativity: (F(x,y) = F(y,x))
2. identity: (F(x, 0) = x = F(0, x))
3. associativity: (F(F(x,y), z) - F(x, F(y,z)) = 0)

We can make sure that our polynomial satisfies these constraints! How? We mod out our coefficient ring (c_{ij}) by the ideal (I) — generated by the relations among (c_{ij}) imposed by these constraints.

If you’d like to see the explicit relations, I wrote a cry for help post on stack overflow.

The ring of coefficients that results is called the Lazard ring (L = \mathbb{Z}[c_{ij}]/I).

It’s important to note here that group-y polynomials are morphisms out of the Lazard ring, not elements of the Lazard ring (i.e., that an assignment of values to each of the (c_{ij}) describes a group-y polynomial, but the ring of the (c_{ij}) itself is just a polynomial ring).

In other words, group-y polynomials (f(x,y)) are morphisms out of the Lazard ring, not elements of the Lazard ring.

More formally: for any ring (R) with group-y polynomial (f(x,y) \in R[[x,y]]) there is a unique morphism (L \to R) that sends (\ell \mapsto f).

(L \to R \simeq F_R)

(where (F_R) denotes a group-y polynomial with coefficients in (R))

This makes sense. If it doesn’t, then scroll up a bit! As we saw above, a change of base ring corresponds to a new group-y polynomial.

Grading the Lazard Ring

As we’ve noted, the Lazard ring (L = \mathbb{Z}[c_{ij}]/I) is the quotient of a polynomial ring on the (c_{ij}) by some relations.

Lazard proved that it is also a polynomial ring (no relations) on a different set of generators. More specifically, (\alpha) is a graded ring isomorphism:

(\mathbb{Z}[c_{ij}]/I \xrightarrow{\alpha} \mathbb{Z}[t_1, t_2 …])

(where the degree of (t_i) is (2i)).

Lurie talks about this a bit (Theorem 4, Lecture 2: The Lazard Ring), but I have yet to understand the proof myself.

Thanks to Alex Mennen for deriving constraints the associativity condition puts on our coefficients; thanks to Qiaochu Yuan and Josh Grochow for kindly explaining some basic details and mechanics of the Lazard ring.

For your ventures ahead…

In this post, I have committed two semantic sins in the name of pedagogy. Namely, sins of oversimplification which I’ll attempt to rectify s.t. you aren’t hopelessly confused by the literature:

1. group-y polynomial = “1-dimensional abelian formal group law”
2. polynomial = “formal power series”

Conventionally, a “polynomial” is a special case of a formal power series (in which we expect that our variables evaluate to a number - useful if we care about convergence).

polynomials (\subset) formal power series

The polynomial ring (R[x]) is the ring of all polynomials (in two variables) over a given coefficient ring (R). The ring of formal power series (R[[x]]) is the ring of all formal power series (in two variables) over a given coefficient ring (R).

polynomial ring (\subset) ring of formal power series R[x] (\subset) R[[x]]

Sources:

Groupes de Lie formels à un paramètre Groupes analytiques en caractéristique 0 Groupes de Lie algébriques (travaux de Chavelley) Formal Group Laws - Ravenel