Related papers: Discriminants and Nonnegative Polynomials

Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether the…

We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by…

We classify entire 2-dimensional area-minimizing or stable surfaces in R^4 with quadratic area growth as algebraic, cut out by a finite union of holomorphic polynomials whose collective degrees are controlled by the density at infinity. As…

We determine conditions on q for the nonexistence of deep holes of the standard Reed-Solomon code of dimension k over F_q generated by polynomials of degree k+d. Our conditions rely on the existence of q-rational points with nonzero,…

We attach two binary codes to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the `Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their…

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$, $n\geq 4$, we consider its higher-order discriminant sets $\{ \tilde{D}_m=0\}$, where $\tilde{D}_m:=$Res$(P,P^{(m)})$, $m=2$, $\ldots$, $n-2$, and their…

We show that for a multivariable polynomial $p(z)=p(z_1, \ldots , z_d)$ with a determinantal representation $$ p(z) = p(0) \det (I_n- K (\oplus_{j=1}^d z_j I_{n_j}))$$ the matrix $K$ is structurally similar to a strictly $J$-contractive…

Given a 0-dimensional scheme in a projective space $\mathbb{P}^n$ over a field $K$, we study the K\"ahler differential algebra $\Omega_{R/K}$ of its homogeneous coordinate ring $R$. Using explicit presentations of the modules…

A classical result of Boole shows that, in characteristic 0, the set of singular degree d hypersurfaces in P^N is a divisor of degree (N+1)(d-1)^N in the projective space of all hypersurfaces. We give here analogous formulae for complete…

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative…

Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[…

In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose…

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…

We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a…

Let $k$ be any field. Let $X \subset \mathbb{P}_k^N$ be a degree $d \geq 2$ hypersurface. Under some conditions, we prove that if $X(K) \neq \emptyset$ for some extension $K/k$ with $n:=[K:k] \geq 2$ and $\gcd(n,d)=1$, then $X(L) \neq…

A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's…

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees…

Let $A$ be an integral domain with quotient field $K$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra. Denote by $D(F)$ the discriminant of a polynomial $F\in A[X]$. Further, given a finite etale algebra $\Omega$,…

In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace…

We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic…