Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 101 / 1

An ideal of definition of a Noetherian local ring $(A,\mathfrak{m})$ is an $\mathfrak{m}$-primary ideal $I$; equivalently, $\sqrt{I}=\mathfrak{m}$, or some power ${\mathfrak{m}}^r$ is contained in $I$.

If $M$ is a finitely generated module of dimension $d$, the Hilbert–Samuel functionagrees for all sufficiently large $n$ with a polynomial of degree $d$. Its leading term iswhere $e_I(M)$ is the Hilbert–Samuel multiplicity.

Assume first that $f\ne 0$ and letbe the least total degree of a nonzero homogeneous part of $f$. The associated graded ring iswhere $f_d$ is the initial homogeneous form. Multiplication by the nonzero polynomial $f_d$ is injective in $k[x,y]$, so the degree-$j$ component has dimensionSumming the components of degrees below $n$ givesThus the Hilbert polynomial isIts leading coefficient is the order of vanishing, or multiplicity, of the plane curve $f=0$ at the origin. If $f=0$, no relation is imposed and $\chi =n(n+1)/2$ for every $n$.

A monomial of weighted degree $n$ is $x^{n-2j}{y}^j$ with $0\le j\le \lfloor n/2\rfloor$. ThereforeThis is a quasipolynomial of period two, not eventually one polynomial: it equals $n/2+1$ for even $n$ and $(n+1)/2$ for odd $n$. The usual eventual-polynomial theorem for a graded algebra assumes a standard graded algebra, generated in degree one. Here the generator $y$ has degree two, so there is no contradiction.