Waring problem for homogeneus forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper I answer this question when the degree of $f$ is greater than the number of variables. To do this I translate the algebraic statement into a geometric one concerning the singularities of linear systems of $P^n$ with assigned singularities.
We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.
The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial $p$ of degree $d$ as a finite sum of $d$-{th} powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any given real binary form $p$ of length at most its degree. In fact, we construct a semialgebraic family of Waring decompositions for $p$. Some examples are shown to highlight the difference between the real and the complex case.
Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients. Firstly, we give an explicit formula for the Waring rank of any binary binomial and several examples to illustrating it. Secondly, we prove that, up to scalar multiplication, there are exactly $\binom{d-1}{2}$ binary forms of degree $d$ with Waring rank two and multiple of three fixed distinct linear forms.
The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We completely classify binary forms of Waring rank 3.
Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.
Waring's problem, of expressing an integer as the sum of powers, has a very long history going back to the 17th century, and the problem has been studied in many different contexts. In this paper we introduce the notion of a Waring subspace and a Waring identifiable subspace with respect to a projective algebraic variety $\mathcal X$. When $\mathcal X$ is the Veronese variety, these subspaces play a fundamental role in the theory of symmetric tensors and are related to the Waring decomposition and Waring identifiability of symmetric tensors (homogeneous polynomials). We give several constructi…
School of Mathematical Sciences, Huaqiao University（华侨大学数学科学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
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 given real binary form with length at most its degree. Secondly, we adapt the Sylvester's Algorithm to the real case in order to determine a Waring decomposition with minimal length and then we establish its real rank. We use bezoutian matrices to achieve a minimal decomposition. We consider all real binary forms of a given degree and we decompose this spac…
A notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the first example of $n,d$, such that the maximum open rank for degree $d$ forms that essentially depend on $n$ variables is strictly greater than the maximum rank. On one hand, the result allows to improve the previously known bounds on open rank, but on the other hand indicates that such bounds are likely quite relaxed. Nevertheless, some of the preparatory resul…
Apolarity is an important tool in commutative algebra and algebraic geometry which studies a form, $f$, by the action of polynomial differential operators on $f$. The quotient of all polynomial differential operators by those which annihilate $f$ is called the apolar algebra of $f$. In general, the apolar algebra of a form is useful for determining its Waring rank, which can be seen as the problem of decomposing the supersymmetric tensor, associated to the form, minimally as a sum of rank one supersymmetric tensors. In this article we study the apolar algebra of a product of linear forms, whic…
We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. Given a word $w$ in a free Lie algebra $\mathcal{L}_{r}$, it induces a word map $φ_{w}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ for every semisimple Lie algebra $\mathfrak{g}$. Given two words $w_{1}\in\mathcal{L}_{r_{1}}$ and $w_{2}\in\mathcal{L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $φ_{w_{1}}*φ_{w_{2}}:=φ_{w_{1}}+φ_{w_{2}}:\mathfrak{g}^{r_{1}+r_{2}}\rightarrow\mathfrak{g}$. We show that …
In this paper, we consider the generalized isomonodromic deformations of rank two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank two irregular connections by apparent singularities. By the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.
We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…
Dynkin diagrams and crepant resolutions of quotient singularities
设$V$是一个复向量空间，有限群$G$通过线性变换作用于其上.令$W = V \oplus V^*$为$V$与其对偶$V^*$的和。我们证明了如果商$W/G$允许光滑的可裂分辨，则子群$G \subset Aut V$是由复反射生成的。我们还得到了关于共轭$W/G$的光滑可裂分解的结构的一些结果，其中$W$是辛向量空间，$G \subset Aut W$是向量空间$W$的辛线性变换的有限群.
In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such a deformation. As a corollary, we show that a deformation of a complex Morse singularity with real linear terms has only indefinite folds and cusps in general and the number of cusps is 3.
We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a linear system of maximal rank. In addition, we show that certain auxiliary parameters can also be fixed. This enables us to conclude that the family of such differential equations is of the expected dimension and to define a birational map between an open subset of the moduli space of loga…
In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$ is nonhyperelliptic. In this article we investigate the situation of normal surface singularities, so we fix a resolution graph $\mathcal{T}$ and a generic singularity with resolution $\tX$ corresponding to it in the sense of \cite{NNII}. We consider an integer effective cycle $Z$ on the resolution $\tX$ and investigate the existence of a complete linear se…
Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then al…
