# An exercise in Linear Algebra

Let $k$ be a field of characteristic $0$ and $k[x_1,\ldots,x_n]^{(d)}$ be the $k$-vector space of homogeneous degree $d$ polynomials in $n$ variables. Show that the linear $k$ span of the set of $d$-th powers of linear polynomials $(a_1 x_1+\ldots+a_n x_n)^d$ is the whole space of homogeneous degree $d$ polynomials.

In coordinate-free terms: if $V$ is a finite-dimensional $k$-vector space then $\{ v^n : v\in V\}$ spans $\mathrm{Sym}^dV$.

# [Some punny title about determinants and fixed points]

Prove the following identity

$\sum_\sigma (-1)^\sigma t^{\mathrm{Fix}\;\sigma} = \det\begin{pmatrix} t & 1 &\cdots& 1& 1\\ 1 & t &\cdots& 1& 1 \\ \vdots & \vdots &\ddots & \vdots & \vdots \\ 1 & 1 &\cdots& t& 1\\ 1 & 1 &\cdots& 1& t \end{pmatrix}$

the sum running through $S_n$ and the matrix being of size $n\times n$.

# Its hip to be skew

Show that the determinant of a skew symmetric $2n\times 2n$ matrix is never reducible when viewed as a polynomial in $\binom {2n} 2$ variables.

# The order of an automorphism

Let $A\in \mathrm{GL}(n, q)$. What is the maximum order $A$ can have?

# Keeping it real

Let $A\in\Bbb{R}^{n\times n}$ be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of $A$ in which all matrices have at least one real eigenvalue.

# A generic curve

Exhibit a curve in $\Bbb{R}^n$ such that any $(n+1)$-uple of points in its trace is in general position.

# Polynomial Decomposition

Decide if there are polynomials $A,B,C,D,E,F$ over $\mathbb{Q}$ such that

$1 + xy + x^2y^2 + x^3y^3 = A(x)B(y) + C(x)D(y) + E(x)F(y)$.