Related papers: Combinatorial Differential Algebra of $x^p$
Let $J_G$ denote the binomial edge ideal of a connected undirected graph on $n$ vertices. This is the ideal generated by the binomials $x_iy_j - x_jy_i, 1\leq i < j \leq n,$ in the polynomial ring $S= K[x_1,...,x_n,y_1,...,y_n]$ where…
We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations,…
This paper concerns the model theory of jet spaces (i.e., higher-order tangent spaces) in differentially closed fields. Suppose p is the generic type of the jet space to a finite dimensional differential-algebraic variety at a generic…
We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable…
The existence of a nondefective stationary disc attached to a nondegenerate model quadric in C^N is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs. In this paper, we give an…
For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the number of these components and their dimensions. When $m$ varies, these components give…
We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…
The geometric motivic Poincar\'e series of a germ $(S,0)$ of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through $(S,0)$. Denef and Loeser proved that this series has a rational…
We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree…
Consider an n-dimensional projective toric variety X defined by a convex lattice polytope P. David Cox introduced the toric residue map given by a collection of n+1 divisors Z_0,...,Z_n on X. In the case when the Z_i are T-invariant…
Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for…
In this follow-up paper, we again inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ or $\in \mathbb{Z}_{p}$ or…
Since the results here have been superseded by another paper cowritten by the author, this article is available for reference purposes only. Fix a Dedekind domain $\mathcal{O}$ and a non-zero prime $\mathfrak{p}$ in it along with a…
We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays…
In this article, we prove a modulo $p$ congruence which connects the class number of the quadratic field $\mathbb{Q}(\sqrt{(-1)^{(p-1)/2}p})$ and the trace of a certain monomial in a root $\theta$ of the Artin-Schreier polynomial…
The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads…
In this article we prove that the adjoint polynomial of arbitrary convex polytopes is up to scaling uniquely determined by vanishing to the right order on the polytopes residual arrangement. This answers a problem posed by Kohn and Ranestad…
A monomial algebra is the quotient of a polynomial algebra by an ideal generated by monomials. We prove that finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with…
In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map…
Let k be an algebraically closed field. We study the cotangent space of a point t corresponding to a monomial ideal I of k[x_1, ..., x_r] in the Hilbert scheme of n points of affine r-space (so the k-dimension of k[x_1, ..., x_r]/I =…