Let $R$ be the set of homogeneous polynomials of degree $n$ in $d$ variables over $\mathbb{C}$. When $n>2$, the set of elements of $R$ that split into a product of linear factors forms a proper subset $S$ of $R$. Is $S$ an algebraic variety, or something almost as nice? If so, how can $S$ be d...

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann-Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable vectors in $\wedge^d V$ (i.e. which are of the form $v_1 \wedge ... \wedge v_d$), thus desc...

Let $\boldsymbol{c}_1, ..., \boldsymbol{c}_n$ be $n$ orthonormal, $m$-dimensional complex vectors, with $\boldsymbol{c}_i = (c_{i,1}, ..., c_{i,m})$. Consider the following polynomial in $x_1,..., x_m$: $$ (c_{11} x_1 + c_{12} x_2 + ... + c_{1m} x_m) (c_{21} x_1 + c_{22} x_2 + ... + c_{2m} x_m) \...

Let $x_1, ..., x_n$ be formal variables. One variant of the Newton-Girard identities expresses $$\sum_{\pi \in S_n} x_{\pi(1)} x_{\pi(2)} \cdots x_{\pi(k)}$$ as a polynomial in the power sums of the $x_i$-s. I am looking for a variant which does the following. For every $1 \le j \le m$, let $x^{...

Let $V: L^2(0,1) \to L^2(0,1)$ be the Volterra integration operator: $V(f)(x) := \int_0^x f(t) \, dt$. Is there a universal function $C(L,\varepsilon) < \infty$ such that the following uniform version of this question holds? For any continuous function $f \in C([0,1])$ with $f(0) = 1$ and $\sup...