## Franklin's Notes

### Polynomials versus forms

We have several powerful results about polynomial equations and their solutions modulo $p$, for primes $p$. A polynomial $F(x_1,\cdots,x_n)$ in $n$ variables over a ring $R$ is a finite linear combination of terms taking the form and the degree of the above term is defined to be $m_1+m_2+\cdots+m_n$. The degree of the polynomial $F$ is defined to be the largest degree of a (nonzero) term of $F$. On the other hand, a form is defined to be a polynomial in which all (nonzero) terms have the same degree. For instance, is a quadratic polynomial, whereas is a quadratic form.