16 problems

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 $n\geq 2$, and define … Let $r(z,\bar z)$ be a real polynomial on the diagonal of $\mathbb{C}^n\times\mathbb{C}^n$ such that … for a holomorphic polynomial mapping $h$. Let…

Let $N\geq 1$. For a collection $H=(H_{i,\alpha}:i\in[N],\alpha\in\mathbb{Z}_{\geq 0}^N,|\alpha|=\delta)$ of complex numbers, let $\mathcal{E}_H(\mathcal{T})$ denote the associated…

Let $f_1,\ldots,f_{n-1}\colon\mathbb C^n\to\mathbb C$ be polynomial functions and write … Let $B(F_{\widehat n})$ denote the set of points at which this mapping is not a locally tr…

Quasi-extremality conjecture. The $T$-symmetrized polynomials $zq(z^T)$ might be quasi-extremal in the Genthner–Ruscheweyh–Salinas sense.

Let $d_1,d_2>0$ and let $\Omega_{\mathbb{C}^2}(d_1,d_2)$ denote the space of polynomial mappings from $\mathbb{C}^2$ with the indicated degrees. Let $U$ be the set of topologically…

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 $d_1,d_2>0$ and let $f\in\Omega(d_1,d_2)$. Write $\Delta(f)$ for the discriminant of $f$; a cusp and a node are the relevant singularities of this discriminant. Conjecture 2. F…

Let $d_1,d_2>0$, let $\Omega(d_1,d_2)$ denote the space of mappings under consideration, and let $U$ be the set of generic mappings in $\Omega(d_1,d_2)$. Conjecture 1. For every…

Let $h_1(x_1,\ldots,x_n),\ldots,h_n(x_1,\ldots,x_n)$ be integral polynomials, and define the polynomial mapping … A polynomial vector is a mapping of this form, and a linear polyno…

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 $z=(z^1,\ldots,z^n)\in\mathbb C^n$. Suppose that $A(z,\bar z)=\lVert F(z)\rVert^2-\lVert G(z)\rVert^2$ for polynomial m…

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…

Let $X$ be a closed semialgebraic set with degree of $cmathbb{R}$-uniruledness at most $d_1$, and let $f:X\rightarrow cmathbb{R}^m$ be a generically finite polynomial mapping of de…

Let $cmathbb{K}$ be an arbitrary algebraically closed field, and let $f:cmathbb{K}^n\rightarrow cmathbb{K}^m$ be a generically finite polynomial mapping of degree $d$. Let…