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.

中文翻译：

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Isabelle/HOL 中代数数的验证实现


我们用 Isabelle/HOL 形式化代数数。我们的开发是对实数和复数代数运算的验证实现。此外，我们还提供了可以识别有理多项式的所有实根或复根的算法，以及两种显示代数数的实现，一种是近似版本，另一种是单射精确版本。我们通过 Isabelle 的代码生成器获得这些操作的经过验证的 Haskell 代码。该开发结合了各种现有的形式化，例如矩阵、Sturm 定理和多项式因式分解，并且包括有关二元多项式、独特因式分解域、结果和子结果的新形式化。