Computing Puiseux series : a fast divide and conquer algorithm

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$.Comment: 27 pages, 2 figure

Computing Puiseux series: a fast divide and conquer algorithm

Let $F ∈ K[X, Y ]$ be a polynomial of total degree D defined over a field K of characteristic zero or greater than D. Assuming F separable with respect to Y , we provide an algorithm that computes all Puiseux series of F above X = 0 in less than $O˜(D δ)$ operations in K, where δ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in $K[[X]][Y ]$ up to an arbitrary precision X N with $O˜(D(δ + N))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with $O˜(D^3)$ arithmetic operations and, if K = Q, with $O˜((h+1) D^3)$ bit operations using probabilistic algorithms, where h is the logarithmic height of F

1974 Warbler

The 1974 Warbler, yearbook of Eastern Illinois Universityhttps://thekeep.eiu.edu/warbler/1053/thumbnail.jp

1976 Warbler

The 1976 Warbler, yearbook of Eastern Illinois Universityhttps://thekeep.eiu.edu/warbler/1058/thumbnail.jp

1976 Warbler

The 1976 Warbler, yearbook of Eastern Illinois Universityhttps://thekeep.eiu.edu/warbler/1058/thumbnail.jp