Jin-San Cheng

On the topology of real algebraic plane curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.

Complete numerical isolation of real zeros in zero-dimensional triangular systems

We present a complete numerical algorithm of isolating all the real zeros of a zero-dimensional triangular polynomial system <i>F</i>_n Z[<i>x</i>₁…,<i>x</i>_n]. Our system <i>F</i>_n is general, with no further assumptions. In particular, our algorithm successfully treat multiple zeros directly in such systems. A key idea is to introduce evaluation bounds and sleeve bounds. We implemented our algorithm and promising experimental results are shown.

On the topology of planar algebraic curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition (CAD) use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.

Determining the topology of real algebraic surfaces

An algorithm is proposed to determine the topology of an implicit real algebraic surface in ℝ3. The algorithm consists of three steps: surface projection, projection curve topology determination and surface patches composition. The algorithm provides a curvilinear wireframe of the surface and the surface patches of the surface determined by the curvilinear wireframe, which have the same topology as the surface. Most of the surface patches are curvilinear polygons. Some examples are used to show that our algorithm is effective.

Root isolation for bivariate polynomial systems with local generic position method

A local generic position method is proposed to isolate the real roots of a bivariate polynomial system ∑={<i>f</i>(<i>x,y</i>),<i>g</i>(<i>x,y</i>)}. In this method, the roots of the system are represented as linear combinations of the roots of two univariate polynomial equations <i>t</i>(<i>x</i>)=0 and <i>T</i>(<i>X</i>)=0: {<i>x</i> = α, <i>y</i> = β -- α/<i>s</i> | α ε <i>V</i>(<i>t</i>(<i>x</i>)), β ε <i>V</i>(<i>T</i>(<i>X</i>)), ||β -- α| < <i>S</i>}, where <i>s</i>, <i>S</i> are constants satisfying certain conditions. The multiplicities of the roots of Σ=0 are the same as that of the corresponding roots of <i>T</i>(<i>X</i>)=0. This representation leads to an efficient and stable algorithm to isolate the real roots of Σ.

Certified rational parametric approximation of real algebraic space curves with local generic position method

In this paper, an algorithm is given for determining the topology of an algebraic space curve and to compute a certified G^1 rational parametric approximation of the algebraic space curve. The algorithm works by extending to dimension one the local generic position method for solving zero-dimensional polynomial equation systems. Here, certified means that the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic position method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, an error bound of the approximation space curve is deduced explicitly from the error bounds of the approximation plane curves. The complexity of the algorithm is also analyzed. Its effectivity is shown on some non-trivial examples.

Root isolation of zero-dimensional polynomial systems with linear univariate representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the complex roots of the polynomial system are represented as linear combinations of the roots of several univariate polynomial equations. An algorithm is proposed to compute such a representation for a given zero-dimensional polynomial equation system based on Grobner basis computation. The main advantage of this representation is that the precision of the roots of the system can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain roots of the polynomial system with a given precision. As a consequence, a root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.

A generic position based method for real root isolation of zero-dimensional polynomial systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is O ? B ( N 10 ) for the bivariate case, where N = max ? ( d , ? ) , d resp., ? is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.

MULTIPLICITY-PRESERVING TRIANGULAR SET DECOMPOSITION OF TWO POLYNOMIALS

In this paper, a multiplicity-preserving triangular set decomposition algorithm is proposed for a system of two polynomials, which involves only computing the primitive polynomial remainder sequence of two polynomials once and certain GCD computations. The algorithm decomposes the unmixed variety defined by two polynomials into square free and disjoint (for non-vertical components, see Definition 4) algebraic cycles represented by triangular sets, which may have negative multiplicities. Thus, the authors can count the multiplicities of the non-vertical components. In the bivariate case, the authors give a complete algorithm to decompose the system into zeros represented by triangular sets with multiplicities. The authors also analyze the complexity of the algorithm in the bivariate case. The authors implement the algorithm and show the effectiveness of the method with extensive experiments.

Topology determination and isolation for implicit plane curves

A method is proposed to generate an isolation for a plane curve, which is a set of boxes covering the curve, having the same topology as the curve, and approximating the curve to any given precision. The method uses symbolic computation to guarantee correctness and uses interval analysis whenever possible to enhance efficiency. This leads to a quite effective hybrid method for plane curve isolation.