相关论文: Exact Symbolic-Numeric Computation of Planar Algeb…

We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the…

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…

We describe new algorithms to compute Whitney stratifications of real algebraic varieties. Using either conormal or polar techniques, these algorithms stratify a complexification of a given real variety. We then show that the resulting…

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and non-local symmetries)…

We present a probabilistic graphical model formulation for the graph clustering problem. This enables to locally represent uncertainty of image partitions by approximate marginal distributions in a mathematically substantiated way, and to…

We study an abstract setting for cutting planes for integer programming called the infinite group problem. In this abstraction, cutting planes are computed via cut generating function that act on the simplex tableau. In this function space,…

Cylindrical algebraic decomposition (CAD) is a core algorithm within Symbolic Computation, particularly for quantifier elimination over the reals and polynomial systems solving more generally. It is now finding increased application as a…

Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…

Given symmetric matrices $A_0, A_1, \ldots, A_n$ of size $m$ with rational entries, the set of real vectors $x = (x_1, \ldots, x_n)$ such that the matrix $A_0 + x_1 A_1 + \cdots + x_n A_n$ has non-negative eigenvalues is called a…

We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise…

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic…

Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gr\"obner bases. This algorithm avoids the use of elimination, homological algebra and processing the…

We present a distributed anytime algorithm for performing MAP inference in graphical models. The problem is formulated as a linear programming relaxation over the edges of a graph. The resulting program has a constraint structure that…

While stabilizer tableaus have proven exceptionally useful as a descriptive tool for additive quantum codes, they offer little guidance for concrete constructions or coding algorithm analysis. We introduce a representation of stabilizer…

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In…

Recent deep learning workloads exhibit dynamic characteristics, leading to the rising adoption of dynamic shape compilers. These compilers can generate efficient kernels for dynamic shape graphs characterized by a fixed graph topology and…

Like [1], we present an algorithm to compute the simulation of a query pattern in a graph of labeled nodes and unlabeled edges. However, our algorithm works on a compressed graph grammar, instead of on the original graph. The speed-up of…

In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential…

Symbolic computation is an important approach in automated program analysis. Most state-of-the-art tools perform symbolic computation as interpreters and directly maintain symbolic data. In this paper, we show that it is feasible, and in…