11 problems

Let $L$ and $\hat{L}$ be the two lattices in the prehomogeneous vector space whose zeta functions are $\xi_i(L,s)$ and $\xi_i(\hat{L},s)$ for $i=1,2,3$, as defined in the paper. Oh…

Let $P_1$, $P_2$, $Q_1$, and $Q_2$ be the relative invariants defined in the previous section. Put … and let $K=\phi_{(r-1,1)}=P_1^{r-1}P_2^{-r^2+2r}$. Conjecture A. The following…

Let $b(\underline{s})$ be a several-variable $b$-function of semi-invariants on a more general prehomogeneous vector space, and suppose it has the form … where $d_i\in\mathbb{N}$,…

Let $X=M_n(\mathbb{C})\times\mathbb{C}^n$, and define … The function $f$ is a relative invariant on $X$. Irreducibility conjecture. The function $f$ is irreducible. This conjecture…

Let $G$ act prehomogeneously on a vector space $V$, and let an additive relative invariant be a rational function associated with an additive function of $G$. Suppose an additive r…

Dolgachev's conjecture. A homaloidal polynomial whose multiplicative Legendre transform is a polynomial must be a relative invariant of a regular prehomogeneous vector space.

Let $(\mathrm{G},V)$ be a strongly prehomogeneous vector space, meaning an algebraic group $\mathrm{G}$ acts linearly on a finite-dimensional vector space $V$ with finitely many or…

Let $(\mathrm{G},V)$ be a prehomogeneous vector space, let $Z\subset V$ be the closure of an orbit of $\mathrm{G}$, and write $Z_{\mathrm{sing}}$ for its singular locus. For an int…

Let $(\mathrm{G},V)$ be a prehomogeneous vector space, meaning that a reductive linear group $\mathrm{G}$ acts on the finite-dimensional vector space $V$ with a dense orbit. Let…

Let $G$ be an algebraic group, let $P$ be a parabolic subgroup with unipotent radical $N$, and let $C$ be a conjugacy class in $P/N$. Let $D$ be the inflation of $C$ to $P$, and le…

Let $V_{\mathbb{Q}}$ be the space of binary cubic forms over $\mathbb{Q}$, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$ act by linear changes of variables, and let $P(x)$ denote th…