平方根的可推广性
Rationalizability of square roots
摘要：
在高能粒子物理理论中，费曼积分计算经常涉及运动变量的平方根。物理学家经常想用多重多项式来求解费曼积分。根据这些函数获得解的一种方法是通过适当的变量变化使所有出现的平方根合理化。本文给出了多项式比的平方根的有理化的严格定义。我们表明，问题的决定是否一个单一的平方根是合理的，可以重新在几何方面。使用这种方法，我们给出了简单的标准，以确定合理性在大多数情况下的平方根在一个和两个变量。我们也给出了部分结果和证明或证伪平方根集合的有理化性的策略。我们将结果应用于高能粒子物理中的许多实际计算实例。
英文摘要：
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of rationalizability for square roots of ratios of polynomials. We show that the problem of deciding whether a single square root is rationalizable can be reformulated in geometrical terms. Using this approach, we give easy criteria to decide rationalizability in most cases of square roots in one and two variables. We also give partial results and strategies to prove or disprove rationalizability of sets of square roots. We apply the results to many examples from actual computations in high energy particle physics.
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