Algebraic Power Series are a lazy representation of multivariate power series with fractional exponents, which are roots of univariate polynomials with coefficients in multivariate polynomial rings. The functionality allows the "lazy" computation of the power series expansion to any finite degree, this being well-determined by the defining algebraic equation.
The package was designed with the computation of formal resolutions of singularities of surfaces in mind but should provide a useful general tool for users. As well as allowing definition directly from a polynomial equation, the user can compose algebraic power series and recursively define series which are roots of polynomials whose coefficients are polynomial functions in other algebraic power series. There are also functions for basic arithmetic operations and tests for exact equality and the like.
The defined series must be expandable in fractional (positive) powers of the base variables. This is true for all roots of a quasi-ordinary polynomial possibly after a finite extension of the base field.
The package was designed and implemented by Tobias Beck at the RICAM institute in Linz, Austria. Some low-level adaptations for added efficiency and integration were carried out by the Magma group. The algorithms are described in [Bec07, Sec. 4].