1 results
On the Dimension of the Locus of Determinantal Hypersurfaces
-
- Journal:
- Canadian Mathematical Bulletin / Volume 60 / Issue 3 / 01 September 2017
- Published online by Cambridge University Press:
- 20 November 2018, pp. 613-630
- Print publication:
- 01 September 2017
-
- Article
-
- You have access
- Export citation
-
The characteristic polynomial
${{P}_{A}}({{x}_{0}},...,{{x}_{r}})$ of an 
$r$-tuple 
$A\,:=({{A}_{1}},...,{{A}_{r}})$ of 
$n\times n$-matrices is defined as1
$${{P}_{A}}({{x}_{0}},...,{{x}_{r}}):=\det ({{x}_{0}}I+{{x}_{1}}{{A}_{1}}+\ldots +{{x}_{r}}{{A}_{r}}).$$We show that if
$r\,\,3$ and $A\,:=({{A}_{1}},...,{{A}_{r}})$ is an $r$-tuple of $n\times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples 
$A'\,:=(A_{1}^{'},...,A_{r}^{'})$ such that 
${{p}_{A}}={{p}_{A'}}$. Equivalently, the locus of determinantal hypersurfaces of degree 
$n$ in 
${{\text{P}}^{r}}$ is irreducible of dimension 
$(r-1){{n}^{2}}+1$.
