Abstract
Polynomials are a central concept to many branches in mathematics and computer science. In particular, manipulation of polynomial expressions can be used to model a wide variety of computation. In this paper, we consider a simple recursive construction of multivariate polynomials over a base ring such as the integers or a (finite) field. We show that this construction allows inductive implementation of polynomial operations such as arithmetic, evaluation, substitution, etc. Furthermore, we can transform a polynomial expression into in a sequence of arithmetic expressions in the base ring and prove the correctness of this transformation in Agda. Combined with our recursive construction, this allows for compiling polynomial expressions over a tower of extension fields into scalar expressions over the ground field, for example. Such a technique is not only interesting in its own right but also finds plentiful application in research areas such as cryptography.
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Notes
- 1.
We use Haskell convention that infix data constructors start with a colon and, for concise typesetting, write \(\mathsf {({:\!\!+})}\) instead of the Agda notation \(\_\) \({:\!\!+}\)\(\_\). In the rest of the paper we also occasionally use Haskell syntax for brevity.
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Acknowledgements
The authors would like to thank the members of IFIP Working Group 2.1 for their valuable comments on the first presentation of this work.
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Cheng, CM., Hsu, RL., Mu, SC. (2018). Functional Pearl: Folding Polynomials of Polynomials. In: Gallagher, J., Sulzmann, M. (eds) Functional and Logic Programming. FLOPS 2018. Lecture Notes in Computer Science(), vol 10818. Springer, Cham. https://doi.org/10.1007/978-3-319-90686-7_5
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DOI: https://doi.org/10.1007/978-3-319-90686-7_5
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