The Role of Benchmarking in Symbolic Computation:(Position Paper)

There is little doubt that, in the minds of most symbolic computation researchers, the ideal paper consists of a problem statement, a new algorithm, a complexity analysis and preferably a few validating examples. There are many such great papers. This paradigm has served computer algebra well for many years, and indeed continues to do so where it is applicable. However, it is much less applicable to sparse problems, where there are many NP-hardness results, or to many problems coming from algebraic geometry, where the worst-case complexity seems to be rare.We argue that, in these cases, the field should take a leaf out of the practices of the SAT-solving community, and adopt systematic benchmarking, and benchmarking contests, as a way measuring (and stimulating) progress. This would involve a change of culture

Digital Collections of Examples in Mathematical Sciences

Some aspects of Computer Algebra (notably Computation Group Theory and Computational Number Theory) have some good databases of examples, typically of the form "all the X up to size n". But most of the others, especially on the polynomial side, are lacking such, despite the utility they have demonstrated in the related fields of SAT and SMT solving. We claim that the field would be enhanced by such community-maintained databases, rather than each author hand-selecting a few, which are often too large or error-prone to print, and therefore difficult for subsequent authors to reproduce.Comment: Presented at 8th European Congress of Mathematician

On fewnomials, integral points and a toric version of Bertini's theorem

An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial g(x)∈ℂ[x] when its square g(x)² has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open. In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations f(x,g(x))=0, where f(x,y) is monic of arbitrary degree in y, and has boundedly many terms in x: we prove that the number of terms of such a g(x) is necessarily bounded. This includes the previous results as extremely special cases. We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus Glm. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of Glm, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials

The Role of Benchmarking in Symbolic Computation:(Position Paper)

There is little doubt that, in the minds of most symbolic computation researchers, the ideal paper consists of a problem statement, a new algorithm, a complexity analysis and preferably a few validating examples. There are many such great papers. This paradigm has served computer algebra well for many years, and indeed continues to do so where it is applicable. However, it is much less applicable to sparse problems, where there are many NP-hardness results, or to many problems coming from algebraic geometry, where the worst-case complexity seems to be rare.We argue that, in these cases, the field should take a leaf out of the practices of the SAT-solving community, and adopt systematic benchmarking, and benchmarking contests, as a way measuring (and stimulating) progress. This would involve a change of culture

Faster Algorithms for Sparse Decomposition and Sparse Series Solutions to Differential Equations

Sparse polynomials are those polynomials with only a few non-zero coefficients relative to their degree. They can appear in practice in polynomial systems as inputs, where the degree of the input sparse polynomial can be exponentially larger than the bit length of the representation of it. This leads to the difficulties when computing with sparse polynomials, as many efficient algorithms for dense polynomials take polynomial-time in the degree, and hence an exponential number of operations in a natural representation of the sparse polynomial. In this thesis, we explore new and faster methods for sparse polynomials and power series. We reconsider algorithms for the sparse perfect power problem and derive a faster sparsity-sensitive algorithm. We then show a fast new algorithm for sparse polynomial decomposition, again sensitive to the sparsity of the input and output. Finally, our algorithms to solve the sparse perfect power and decomposition problems lead us to explore a generalization to solving the linear differential equation with sparse polynomial coefficients using a Newton-like method. We demonstrate an algorithm which will find sparse solutions if they exist, in time polynomial in the input and the output

Enabling Machine Science through Distributed Human Computing

Distributed human computing techniques have been shown to be effective ways of accessing the problem-solving capabilities of a large group of anonymous individuals over the World Wide Web. They have been successfully applied to such diverse domains as computer security, biology and astronomy. The success of distributed human computing in various domains suggests that it can be utilized for complex collaborative problem solving. Thus it could be used for machine science : utilizing machines to facilitate the vetting of disparate human hypotheses for solving scientific and engineering problems. In this thesis, we show that machine science is possible through distributed human computing methods for some tasks. By enabling anonymous individuals to collaborate in a way that parallels the scientific method -- suggesting hypotheses, testing and then communicating them for vetting by other participants -- we demonstrate that a crowd can together define robot control strategies, design robot morphologies capable of fast-forward locomotion and contribute features to machine learning models for residential electric energy usage. We also introduce a new methodology for empowering a fully automated robot design system by seeding it with intuitions distilled from the crowd. Our findings suggest that increasingly large, diverse and complex collaborations that combine people and machines in the right way may enable problem solving in a wide range of fields