Monomie

2変数多項式の位相幾何学的情報を計算する

ソフト詳細説明

2変数多項式の位相幾何学的情報を計算するソフト

　In recent years there has been much interest in the study of degenerations of algebraic curves.　When curves are rational or elliptic, Kodaira [K] completes the classification of degenerations.　When general fibers have a larger genus, classification problem is so hard to solve.　Matsumoto and Montesinos [MM] show that classification of degenerations is equivalent to classification of pseudo-periodic homeomorphisms.　It follows that we have a topological solution for this problem.

　\monomie ('Monomii-kun' in Japanese) is a computer software which runs on only Macintosh.　It was planned to observe classification of singular fibers and their monodromies for any given fibrations.　The latest version is \monoversion and in this version we can know topological data of given algebraic curves in ${\bf C}^2$ even if they have singularities.

　We mention approximating problem of roots of algebraic equations.　The most important parts of the program of \monomie is those about this problem.　In other words, \monomie is a software to 'get roots' of an equation with 2 variables.　It is not so easy to calculate multiple roots precisely in case of algebraic equations, and in \monomie we can not avoid the difficulty.　Because in the prime step, \monomie gets an equation $F_t(x)=0$ of $x$ with one parameter $t$ and computes parameters such that the equation has multiple roots.　Even if we obtain such a parameter $t\sim t_0$ approximately, $F_{t_0}(x)=0$ may has no multiple solutions because $t_0$ is not a explicit value.　In easy (less than 10 degree) case \monomie overcomes this trouble and gives us topological data.

　finally, we have an application of \monomie to complex singularities.　Suppose that an algebraic curve in $\cc^2$ has a discrete singular point $p$.　Around $p$ there happens a cone of a link generally.　\monomie can calculate the link type of this cone if the curve is given by a polynomial.