相关论文: Codimension one decompositions and Chow varieties

We review work on `decomposition,' a property of two-dimensional theories with 1-form symmetries and, more generally, d-dimensional theories with (d-1)-form symmetries. Decomposition is the observation that such quantum field theories are…

A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…

The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree $d$ in $n+1$ variables on an algebraically closed field, called $\Split_{d}(\PP n)$, with the Grassmannian of $n-1$…

We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then based on the generic intersection theory, we define the partial differential Chow form for an irreducible partial…

We develop a unified mathematical theory of defect condensations for topological orders in all dimensions based on higher categories, higher algebras and higher representations. A k-codimensional topological defect $A$ in an n+1D…

We examine the five-dimensional super-de Rham complex with $N = 1$ supersymmetry. The elements of this complex are presented explicitly and related to those of the six-dimensional complex in $N = (1, 0)$ superspace through a specific notion…

The Waring Problem over polynomial rings asks for how to decompose an homogeneous polynomial of degree $d$ as a finite sum of $d^{th}$ powers of linear forms. First, we give a constructive method to obtain a real Waring decomposition of any…

In this paper, a generic intersection theorem in projective differential algebraic geometry is presented. Precisely, the intersection of an irreducible projective differential variety of dimension d>0 and order h with a generic projective…

Using Lipman's results on resolution of two-dimensional singularities, we provide a form of resolution of singularities in codimension two for reduced quasi-excellent schemes. We deduce that operations of degree less than two on algebraic…

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which…

Two new concepts, generic regular decomposition and regular-decomposition-unstable (RDU) variety for generic zero-dimensional systems, are introduced in this paper and an algorithm is proposed for computing a generic regular decomposition…

The cosimplicial scheme $$Delta^bullet = \Delta^0 smallmatrix \to smallmatrix \Delta^1 smallmatrix to smallmatrix ...;\quad \Delta^n :=\Spec\Big(k[t_0,...c,t_n]/(\sum t_i -t)\Big)$$ was used in B to define higher Chow groups. In this note,…

A variation of Waring's problem from classical number theory is the question, ``What is the smallest number $s$ such that any generic homogeneous polynomial of degree $d$ in $n+1$ variables may be written as the sum of at most $s$ products…

In this paper we discuss gauging one-form symmetries in two-dimensional theories. The existence of a global one-form symmetry in two dimensions typically signals a violation of cluster decomposition -- an issue resolved by the observation…

Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…

The space of codimension one holomorphic foliations of degree 1 in a projective space has an irreducible component whose general element is a logarithmic differential 1-form with simple poles in three hyperplanes. We compute its projective…

Consider an ideal $I \subset R = \bC[x_1,...,x_n]$ defining a complex affine variety $X \subset \bC^n$. We describe the components associated to $I$ by means of {\em numerical primary decomposition} (NPD). The method is based on the…

The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…

We present a generalization of first-order unification to a term algebra where variable indexing is part of the object language. We exploit variable indexing by associating some sequences of variables ($X_0,\ X_1,\ X_2,\dots$) with a…

This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the…