degdenominator

Degdenominator is a mathematical invariant used in the study of Puiseux series and other fractional-power representations. It serves to quantify the denominators that appear in the exponents of a generalized power series and helps standardize manipulations of such series.

Formally, if a function is expressed as a Puiseux series f(x) = sum a_r x^r with exponents r in Q and only finitely many distinct denominators occur, the degdenominator is defined as the least common multiple of the denominators of those exponents when written in lowest terms. This value indicates how far one must reparameterize the variable to obtain an ordinary power series with integer exponents.

Example: Consider f(x) = x^{1/2} + x^{3/4}. The denominators are 2 and 4, so the degdenominator is lcm(2, 4) = 4. With t = x^{1/4}, the function becomes f(t^4) = t^2 + t^3, a standard power series in t with integer exponents.

Computation involves listing the exponents, reducing them to lowest terms, and taking the least common multiple of their denominators. Applications include simplifying algebraic and analytic manipulations of fractional power series, aiding Ramification and singularity analysis, and improving efficiency in computer algebra systems when handling Puiseux expansions.

Limitations: the concept applies to finite sums or convergent Puiseux expansions with rational exponents; it is not defined for series containing irrational exponents or infinitely many distinct denominators in a problematic sense.

See also: Puiseux series, fractional power series, ramification index.