Abstract
We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.
Highlights
IntroductionI.e., the numbers that are expressed as roots of non-zero integer (or equivalently rational) polynomials, are an attractive subset of the real or complex numbers
Algebraic numbers, i.e., the numbers that are expressed as roots of non-zero integer polynomials, are an attractive subset of the real or complex numbers
For Isabelle, Li and Paulson [18] independently implemented algebraic numbers. They did not formalize resultants; instead, they employed an external tool as an oracle to provide polynomials that represent desired algebraic numbers, and provided a method to validate that the polynomials from the oracle are suitable
Summary
I.e., the numbers that are expressed as roots of non-zero integer (or equivalently rational) polynomials, are an attractive subset of the real or complex numbers. Most of the algorithms and proofs of our formalization are based on a textbook by Mishra [20, Chapters 7 and 8]; it contains a detailed implementation of real algebraic numbers, including proofs. When it comes to subresultants, we followed the original papers by Brown and Traub [2,3]. For Isabelle, Li and Paulson [18] independently implemented algebraic numbers They did not formalize resultants; instead, they employed an external tool as an oracle to provide polynomials that represent desired algebraic numbers, and provided a method to validate that the polynomials from the oracle are suitable..
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