6 problems

Existence conjecture. Does there exist a function $\rho$ such that

Fix an integer $d\geq 3$. For each $n\geq 1$, let $H=(H_{i,\alpha}:i\in[n],\lvert\alpha\rvert=d)$ be a collection of complex numbers, and let $\Psi_{i,j}^{\alpha}(H)$ and…

Let $F=(f_1,\ldots,f_n):\mathbb R^n\to\mathbb R^n$ be a polynomial mapping. Write $J(F)$ for its Jacobian determinant and $S_F$ for its non-properness set, and let…

Let $G=(G_1,\ldots,G_{n-1}):\mathbb{C}^n\to\mathbb{C}^{n-1}$ be a polynomial mapping with $n\geq 2$ and $K_0(G)=\emptyset$. Assume that there exists a very good projection with res…

Let $F:\mathbb{C}^n\to\mathbb{C}^n$ be a dominant polynomial mapping. Write $K_0(F)$ for the set of critical values of $F$ and $S_F$ for its asymptotic set. Pure-dimensionality con…

Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots,z^n)\in\mathbb C^n$, and let $A(z,\bar z)$ be a Hermitian polynomial satisfying … If $r$ denotes the lin…