论文标题
使用限制集方法严格地求解具有深度2的参数激进方程
Solving parametric radical equations with depth 2 rigorously using the restriction set method
论文作者
论文摘要
我们回顾了关于自由基方程的历史和以前的文献，并介绍了深度2的自由基方程的严格解决方案理论，继续对深度1的自由基方程进行了先前的研究。深度2的自由基方程是方程式的方程式，其中未知变量至少在一个平方根下出现，并且需要两个步骤来消除所有在方程中出现的激进分子。我们指出并证明所有三个方程式的定理具有深度2，从而提供了所有实价解决方案的解决方案集。通过使用不平等限制来决定是接受还是拒绝候选解决方案的限制集方法显示了定理。我们区分正式的解决方案，这些解决方案在正式的意义上满足原始方程式，在这种意义上，我们允许一些激进分子在验证过程中评估对假想数的数字，以及在验证过程中所有激进分子在验证过程中对实数进行评估。我们的定理明确确定了所有方程式形式的所有正式解决方案的集合以及所有强解决方案的集合。具有深度2的自由度方程的基础理论比具有深度1的自由基方程的理论更丰富，更有趣，并且该理论的某些方面在直觉上并不明显。用参数激进方程的示例说明了这一点。
We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to eliminate all radicals appearing in the equation. We state and prove theorems for all three equation forms with depth 2 that give the solution set of all real-valued solutions. The theorems are shown via the restriction set method that uses inequality restrictions to decide whether to accept or reject candidate solutions. We distinguish between formal solutions that satisfy the original equation in a formal sense, where we allow some radicals to evaluate to imaginary numbers during verification, and strong solutions, where all radicals evaluate to real numbers during verification. Our theorems explicitly identify the set of all formal solutions and the set of all strong solutions for each equation form. The theory underlying radical equations with depth 2 is richer and more interesting than the theory governing radical equations with depth 1, and some aspects of the theory are not intuitively obvious. It is illustrated with examples of parametric radical equations.
