# mathlibdocumentation

ring_theory.witt_vector.witt_polynomial

# Witt polynomials #

To endow witt_vector p R with a ring structure, we need to study the so-called Witt polynomials.

Fix a base value p : ℕ. The p-adic Witt polynomials are an infinite family of polynomials indexed by a natural number n, taking values in an arbitrary ring R. The variables of these polynomials are represented by natural numbers. The variable set of the nth Witt polynomial contains at most n+1 elements {0, ..., n}, with exactly these variables when R has characteristic 0.

These polynomials are used to define the addition and multiplication operators on the type of Witt vectors. (While this type itself is not complicated, the ring operations are what make it interesting.)

When the base p is invertible in R, the p-adic Witt polynomials form a basis for mv_polynomial ℕ R, equivalent to the standard basis.

## Main declarations #

• witt_polynomial p R n: the n-th Witt polynomial, viewed as polynomial over the ring R
• X_in_terms_of_W p R n: if p is invertible, the polynomial X n is contained in the subalgebra generated by the Witt polynomials. X_in_terms_of_W p R n is the explicit polynomial, which upon being bound to the Witt polynomials yields X n.
• bind₁_witt_polynomial_X_in_terms_of_W: the proof of the claim that bind₁ (X_in_terms_of_W p R) (W_ R n) = X n
• bind₁_X_in_terms_of_W_witt_polynomial: the converse of the above statement

## Notation #

In this file we use the following notation

• p is a natural number, typically assumed to be prime.
• R and S are commutative rings
• W n (and W_ R n when the ring needs to be explicit) denotes the nth Witt polynomial