**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 coefficent 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

- $F(x,y) = F(y,x)$ commutativity
- $F(x, 0) = x = F(0, x)$ identity
- $F(F(x,y), z) – F(x, F(y,z)) = 0$ associativity

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 amoung $c_{ij}$ 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:

- group-y polynomial = “1-dimensional abelian formal group law”
- 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