Difference algebra

Remarks

Some authors require $\sigma$ to be an automorphism in the definition of a difference ring or field. A difference polynomial over $(R, \sigma)$ is a polynomial in the variables $y, \sigma y, \sigma^2 y, \dots$, and a solution is an element $a$ of a difference ring extension at which it vanishes.

Example: $(\C(x), \sigma)$ with $\sigma(f)(x) = f(x + 1)$ is an inversive difference field. The shift is additive and multiplicative, it fixes $1$, and its inverse is $f(x) \mapsto f(x - 1)$.

Example: a field $K$ of characteristic $p > 0$ with the Frobenius map $\sigma(a) = a^p$ is a difference field.

Non-example: $(\Z, \sigma)$ with $\sigma(n) = 2n$ is not a difference ring. It satisfies clause 1, but $\sigma(1) = 2 \ne 1$, so clause 3 fails, and $\sigma(1 \cdot 1) = 2 \ne 4 = \sigma(1)\sigma(1)$, so clause 2 fails too.