This paper derives an explicit formula for a type of fractional power series, known as a Puiseux series, arising in a wide class of applied problems in the physical sciences and engineering. Detailed consideration is given to the gaps which occur in these series (lacunae); they are shown to be determined by a number-theoretic argument involving the greatest common divisor of a set of exponents appearing in the Newton polytope of the problem, and by two number-theoretic objects, called here Sylvester sets, which are complements of Frobenius sets. A key tool is Faà di Bruno’s formula for high derivatives, as implemented by Bell polynomials. Full account is taken of repeated roots, of arbitrary multiplicity, in the leading-order polynomial which determines a fractional-power expansion, namely the facet polynomial. For high multiplicity, the fractional powers are shown to have large denominators and contain irregularly spaced gaps. The orientation and methods of the paper are those of applications, but in a concluding section we draw attention to a more abstract approach, which is beyond the scope of the paper.

本文推导了一种类型的分数次幂级数的明确公式，称为Puiseux级数，它在物理科学和工程学中的广泛应用问题中产生。详细考虑了在这些系列（缺陷）中出现的差距；它们被证明是由涉及该问题的牛顿多面体中出现的一组指数的最大公约数的数论论证，以及由两个数论对象（在此称为Sylvester集）确定的，这些对象是Frobenius集的补充。关键工具是由贝尔多项式实现的Faàdi Bruno高阶导数公式。在确定分数次幂扩展的前导多项式（即构面多项式）中，充分考虑了任意多重性的重复根。对于高多样性，分数次方具有大的分母，并且包含不规则间隔的间隙。本文的方向和方法是针对应用程序的，但是在最后一节中，我们提请注意一种更抽象的方法，这超出了本文的范围。