Browsing by Author "Song, Ning"

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    A computational method for constructing Sylvester-style sparse resultants
    (2005) Song, Ning; Goldman, Ron
    We present a computational approach for constructing Sylvester style resultants for sparse systems of bivariate polynomial equations. Necessary and sufficient conditions are derived which guarantee that a multiplying set of monomials generates an exact Sylvester style resultant for three bivariate polynomials with a given planar Newton polygon. These conditions include a set of Diophantine equations that can be solved to generate multiplying sets of monomials and therefore the corresponding Sylvester resultants. We have implemented this method in Mathematica, and the results show that such Sylvester style sparse resultants often exist, and they appear in certain specific patterns. This method of Diophantine equations can also be used together with moving planes and moving quadrics [16, 17] to find the implicit equation of a rational surface. Moving planes and moving quadrics were originally introduced for tensor product surfaces---that is, bivariate polynomial systems whose Newton polygons are rectangles. Now by a method similar to our technique for generating Sylvester style sparse resultants, we can use moving quadrics to generate implicit equations for certain rational parametric surfaces whose Newton polygons are not rectangles.
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    Mu -bases and their applications in geometric modeling
    (2007) Song, Ning; Goldman, Ronald
    This thesis defines the notion of a μ-basis for an arbitrary number of polynomials in one variable. The properties of these μ-bases are derived, and a straightforward algorithm is provided to calculate a μ-basis for any collection of univariate polynomials. Systems where base points are present are also discussed. μ-bases are then applied to solve implicitization, inversion and intersection problems for rational space curves. Next, a natural one to one correspondence is derived between the singular points of rational planar curves and the axial moving lines that follow these curves. This correspondence is applied together with μ-bases to compute and to analyze all the singular points of low degree rational planar curves.