We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex-and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 18 to 100 times. We implement our algorithm using the experimental CGAL package triangulation. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. Our method computes instances of 5-, 6-or 7-dimensional polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Compared to tropical geometry software, ours is faster up to dimension 5 or 6, and competitive in higher dimensions.

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. Fig. 1 a Curve of equation 16x 5 − 20x 3 + 5x − 4y 3 + 3y = 0 plotted in maple, b its isotopic graph computed by isotop, and c their overlay

International audienceWe 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 Groebner 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 nongeneric curves