Math 380, Spring 2018, Assignment 14

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

Solve the following problems:

1. Working in $\mathbb{R}^2$, consider the variety $V=\mathbb{V}(x^2y^2-1)$.

(a) Make a sketch of $V$. What do you think its "dimension" ought to be?

(b) Compute a Grőbner basis for the ideal $I=\left\langle x^2y^2-1\right\rangle$.

(c) Make a diagram of $\left\langle\mathrm{LT}(I)\right\rangle$, showing the monomials of this ideal as integer points in the plane.

(d) For each non-negative integer $d$, let $H(d)$ denote the number of monomials of total degree $d$ or less that lie outside $\left\langle\mathrm{LT}(I)\right\rangle$. Compute $H(d)$ for $0\leq d\leq 10$. (The function $H(d)$ is called the Hilbert function associated with $I$.)

(e) Make a graph of $H(d)$ as a function of the integer $d$. Show that, although there is no line passing through all the points of this graph, there is a line that passes through all its points for all sufficiently large $d$. That is, show that $H(d)$ is eventually linear.

2. Repeat the problem above, replacing $V$ by the variety $\mathbb{V}(x^2-1, y^2-1)$. Show in the end that in this case the Hilbert function is eventually constant.

3. (Optional) Devise an algorithm that loops through the leading terms of a Grőbner basis of an ideal, and decides whether its Hilbert function is eventually constant. In other words, devise an algorithm that decides whether a given variety is finite.

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