Homework 4

This homework covers the material from lectures 14 to 17

Due date: March 22nd.

LaTeX template for homework 4, in case you want to write it in LaTeX.

Required Exercises

Remember that you are only required to turn in 5 out of the 6 exercises below.

Problem 1 - Polynomial Parametric Curves in Algebraic Sets - CLO exercise 2.1.5 (20 points)

1. Show that the number of distinct monomials $x^a y^b$ of total degree $\leq m$ in $\mathbb{F}[x,y]$ is equal to $(m+1)(m+2)/2$.
2. Show that if $f(t)$ and $g(t)$ are polynomials of degree $\leq n$ in $t$, then for $m$ large enough, the “monomials” $$ f(t)^a g(t)^b $$ with $a+b \leq m$ are linearly dependent.
3. Deduce from part (2) that if $C$ is a curve in $\mathbb{F}^2$ given parametrically by $x = f(t), \ y = g(t)$ for $f(t), g(t) \in \mathbb{F}[t]$, then $C$ is contained in $V(F)$ for some non-zero $F \in \mathbb{F}[x,y]$.
4. Generalize parts (1), (2) and (3) to show that any polynomial parametric surface $$ x = f(t,u), \ \ y = g(t,u), \ \ z = h(t,u) $$ is contained in an algebraic surface $V(F)$, where $F \in \mathbb{F}[x,y,z]$ is non-zero.

Problem 2 - Exercises on S-polynomials - from CLO 3.6 (20 points)

1. Compute $S(f, g)$ using the lex order for:
   • $f = 4x^2z - 7y^2, \ g = xyz^2 + 3xz^4$
   • $f = x^4y - z^2, \ g = 3xz^2 -y$
2. Does $S(f,g)$ depend on the monomial order being used? Illustrate your assertion with examples.
3. Prove that $multideg(S(f, g)) < \gamma$, where $x^\gamma = LCM(LM(f), LM(g))$. Explain why this inequality is a precise version of the claim that S-polynomials are designed to produce cancellation.
4. Let $f, g \in \mathbb{F}[x_1, \ldots, x_n]$ be nonzero and $x^\alpha, x^\beta$ be monomials. Verify that

$$ S(x^\alpha f, x^\beta g) = x^\gamma S(f,g) $$

where

$$ x^\gamma = \dfrac{LCM(x^\alpha LM(f), x^\beta LM(g))}{LCM(LM(f), LM(g))}. $$

Problem 3 - Exploring Groebner bases - CLO 2.6 and 2.7 (20 points)

1. Let $I \subseteq \mathbb{F}[x_1, \ldots, x_n]$ be an ideal, and let $G$ be a Groebner basis of $I$.

• Show that $f^G = g^G$ if, and only if, $f-g \in I$.
• Show that $(f+g)^G = f^G + g^G$
• Deduce that $(fg)^G = (f^G \cdot g^G)^G$

2. Let $G$ and $G'$ be Groebner bases for an ideal $I$ with respect to the same monomial order in $\mathbb{F}[x_1, \ldots, x_n]$. Show that $f^G = f^{G’}$ for any $f \in \mathbb{F}[x_1, \ldots, x_n]$. Hence, the remainder on division by a Groebner basis is even independent of which Groebner basis we use, as long as we use the same monomial ordering.
3. Show that $\{ y - x^2, z - x^3 \}$ is not a Groebner basis for $(y - x^2, z - x^3)$ with lex order with $x > y > z$

Problem 4 - Groebner bases and Geometry - CLO 2.8 and 2.9 (20 points)

1. Groebner basis and Lagrange interpolation: let $V = \{ (a_1, b_1), \ldots, (a_n, b_n) \} \subset \mathbb{F}^2$, where $a_1, \ldots, a_n$ are distinct. The Lagrange interpolation polynomial of $V$ is given by:

$$ h(x) = \sum_{i=1}^n b_i \cdot \prod_{j \neq i} \dfrac{x-a_j}{a_i - a_j} \in \mathbb{F}[ x ] $$

We will now see how $h(x)$ relates to the reduced Groebner basis of $I(V) \subseteq \mathbb{F}[x, y]$.

• Prove that $I(V) = (f(x), y - h(x))$, where $f(x) = \prod_{i=1}^n (x-a_i)$.

Hint: divide $g \in I(V)$ by $f(x), y-h(x)$ using lex order with $y > x$.

• Prove that $\{ f(x), y-h(x) \}$ is the reduced Groebner basis for $I(V) \subseteq \mathbb{F}[x,y]$ for lex order with $y > x$.

2. A critical point of a differentiable function $f(x,y)$ is a point where all the partial derivatives $\partial_x f$ and $\partial_y f$ vanish simultanously. When $f \in \mathbb{R}[x,y]$, it follows that the critical points can be found by applying our techniques to the system of polynomial equations

$$ \partial_x f = \partial_y f = 0 $$

Consider the function:

$$ f(x,y) = (x^2+ y^2-4)(x^2 + y^2 -1) + (x-3/2)^2 + (y-3/2)^2 $$

• Find all critical points of $f$
• Classify your critical points as local maxima, local minima, or saddle points.

Hint: use the second derivative test.

• If you didn’t do it before, compute the Groebner basis of the ideal $(\partial_x f , \partial_y f)$

Problem 5 - Computing the invariant ring for the left multiplication action (20 points)

In this exercise we will work out how to prove that the invariant ring for the left multiplication action $G = \mathbb{SL}(n)$ on matrices $V = Mat(n)$ is given by $\mathbb{C}[ X ]^G = \mathbb{C}[\det(X)]$, $X = (x_{i,j})_{i,j=1}^n$.

Let $I$ be the ideal of $\mathbb{C}[X]$ generated by all non-constant, homogeneous invariant polynomials.

1. Prove that if a matrix $A \in V$ is singular, then any non-constant homogeneous invariants $p \in I$ must vanish on $A$.

Hint: to show the above, show that for a singular matrix there is a sequence of group elements $\{ g_k \}_{k \geq 1}$ such that $\lim_{k \rightarrow \infty} (g_k \cdot A) = 0$. Then use the fact that polynomials are continuous functions to prove that if $p$ is a homogeneous invariant then $p(A)$ must be equal to zero.

Hint to hint: think of row reduction.

2. Conclude that the zero set of all polynomials in $I$ is the set of singular matrices.

Hint: show that a non-singular matrix $B$ is not in the zero set of the polynomials in $I$ by finding one polynomial where $B$ does not evaluate to zero.

3. Conclude that $I = ( \det(X) )$.

4. Since $\det(X)$ is an irreducible polynomial, we showed that $I$ is a prime ideal. From this, conclude that any homogeneous non-constant invariant must be a power of $\det(X)$. Then conclude that $\mathbb{C}[X]^G = \mathbb{C}[\det(X)]$

Problem 6 - Bivariate Factorisation (20 points)

Factor the following polynomial using the bivariate factorisation algorithm from class.

$$p(x,y) = 5x^5y + 2x^4y^2 + 6x^3y^3 - x^2y^4 + 5x^4y - 5x^3y^2 - 3x^2y^3 + 4xy^4 - 7x^3 - x^2y + 8xy^2 + 4y^3 - 6x^2 + 2xy - 3y^2 $$

Where $p(x,y) \in \mathbb{Z}_{17}[x, y]$. You can, and are highly encouraged to, use Macaulay 2 to handle the steps of the algorithm.

Practice Problems

You are not required to submit the solution to these problems, but highly encouraged to attempt them for your own knowledge sake. :)

Any problem from CLO Chapters 2 and 3.

1. Do exercise 13 of CLO 2.10 to understand how different monomial orderings can lead to very different bases.

2. Do exercises 12 and 13 from CLO 3.6 to see when the resultant does not behave nicely with respect to specializations.

3. Exercise 14 from CLO 3.6 to prove property of Resultant under variable substitution.