Related papers: On the intrinsic complexity of elimination problem…
Probabilistic circuits (PCs) are a class of tractable probabilistic models that allow efficient, often linear-time, inference of queries such as marginals and most probable explanations (MPE). However, marginal MAP, which is central to many…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding…
This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…
We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…
We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of first-order functional programs. We explain their semantics and prove that they form a…
Implementations of Roetteler's shifted bent function algorithm have in recent years been used to test and benchmark both classical simulation algorithms and quantum hardware. These circuits have many favorable properties, including a…
Most classical results in circuit complexity theory concern circuits over the Boolean domain. Besides their simplicity and the ease of comparing different languages, the actual architecture of computers is also an important motivating…
Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the…
Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
Motivated by algorithmic problems from combinatorial group theory we study computational properties of integers equipped with binary operations +, -, z = x 2^y, z = x 2^{-y} (the former two are partial) and predicates < and =. Notice that…
Many computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the "curse of dimensionality", an exponential cost…
So far, the scope of computer algebra has been needlessly restricted to exact algebraic methods. Its possible extension to approximate analytical methods is discussed. The entangled roles of functional analysis and symbolic programming,…
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We…
We explain how to encode an algebraic series by finite data and how to do effective arithmetic on the level of these encodings. The reasoning is based on the Newton-Puiseux algorithm and an effective equality test for algebraic series.…