标题： 三次项不定方程 显示英文标题

标题： Diophantine equations with three monomials

摘要： 我们提出一个通用算法，用于求解所有由三个单项式组成的二元多项式不定方程。在此工作之前，甚至对于包含参数的方程族$x^4+axy+y^3=0$的算法存在性仍是一个开放问题。我们还提出一种基本方法，将寻找一般三单项式方程的所有整数解的任务转化为寻找在不同变量中的三单项式方程的原始解的任务。我们确定了一个大的三单项式方程类，该方法对此类方程可以得到完整的解。经验数据表明，随着变量数量趋于无穷大，此类方程包含$100\%$的三单项式方程。 显示英文摘要

摘要： We present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations $x^4+axy+y^3=0$ has been an open question. We also present an elementary method that reduces the task of finding all integer solutions to a general three-monomial equation to the task of finding primitive solutions to equations with three monomials in disjoint variables. We identify a large class of three-monomial equations for which this method leads to a complete solution. Empirical data suggests that this class contains $100\%$ of three-monomial equations as the number of variables goes to infinity.