Root neighborhoods of a polynomial
Mathematics of Computation
On the numerical condition of polynomials in Berstein form
Computer Aided Geometric Design
Hierarchical Data Structures and Algorithms for Computer Graphics. Part I.
IEEE Computer Graphics and Applications
Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
On the optimal stability of the Bernstein basis
Mathematics of Computation
A fast technique for the generation of the spectral set of a polytope of polynomials
Automatica (Journal of IFAC)
ACM Transactions on Mathematical Software (TOMS)
Bipolar and Multipolar Coordinates
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Pseudozeros of multivariate polynomials
Mathematics of Computation
Minkowski Roots of Complex Sets
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
Computing the Minkowski Value of the Exponential Function over a Complex Disk
Computer Mathematics
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A root neighborhood (or pseudozero set) of a degree-n polynomial p(z) is the set of roots of all polynomials P˜(z) whose coefficients differ from those of p(z), under a specified norm in Cn+1, by no more than a given amount ε. In the case of a weighted infinity norm, root neighborhoods are bounded by generalized lemniscates (algebraic curves of degree 4n). A simple description of generalized lemniscates may be cast in terms of multipolar coordinates: namely, the product of the distances r1,...,rn of a variable point z from the n roots z1,...,zn of p(z) is equal to a degree-n polynomial in the distance r of z from the origin (in the case of "ordinary" lemniscates, this polynomial specializes to a constant). The ability to efficiently and faithfully graph lemniscates is important in numerical analysis, control theory, and other scientific/engineering applications. By constructing the tensor-product Bernstein form over a bounding rectangle, we develop a robust adaptive algorithm to plot lemniscates, employing quadtree subdivision to efficiently achieve any desired resolution. Moreover, singular points of the lemniscate (if any) may be explicitly identified and incorporated into the subdivision.