Show that the ideal of the image of the map $t\mapsto(t,t^3,t^5)$ is given by $I=(y-x^3,z-x^2y)$.

Question

Consider the map $f:\mathbb{A}^1\to \mathbb{A}^3$ with $f(t)=(t,t^3,t^5)$. I want to show that $I(f(\mathbb{A}))=(y-x^3,z-x^2y)$.

First, I can show that $(y-x^3,z-x^2y)\subset I(X)$ where $X$ is the image of $f$. But, I don't know how to show another inclusion that $I(X)\subset(y-x^3,z-x^2y)$.

What is general approach in finding the generators of the ideal of the image of $f$ where $f$ is the map $t\mapsto(t^i,t^j,t^k)$ with $i\leq j\leq k$?

You seem to be asking two different questions: whether the image is closed and whether the ideal of the image is the one you guessed. To answer the former it is not necessary to answer the latter. — Zhen Lin, Oct 15 '20 at 01:11
@ZhenLin, okay, I did some edits. I am basically interested in finding the ideal of the image of $f$. Thanks! — user89898989, Oct 15 '20 at 01:21
This is an example of implicitization and can be solved using Gröbner bases. This post is related. — Viktor Vaughn, Oct 15 '20 at 01:27

Avi Steiner · Answer 1 · 2020-10-15T13:39:59.773

1

This is a special case of a general setup for toric ideals. In this case, here’s your answer:

Let $A$ be the 1-by-3 matrix $[i, j, k]$. Then the ideal you're looking for is $$\left(\prod_{u_l> 0}x_l^{u_l}- \prod_{u_l<0} x_l^{-u_l} : \mathbf{u}\in \mathbb{Z}^3, A\mathbf{u}=0\right). $$

edited Oct 15 '20 at 13:39

answered Oct 15 '20 at 03:41

Avi Steiner

4,209

Show that the ideal of the image of the map $t\mapsto(t,t^3,t^5)$ is given by $I=(y-x^3,z-x^2y)$.

1 Answers1