In this paper, we try to clarify some of the questions related to a key concept in multivariate polynomial solving algorithm over a finite field: the degree of regularity. By the degree of regularity, here we refer to a concept first presented by Dubois and Gama, namely the lowest degree at which certain nontrivial degree drop of a polynomial system occurs. Currently, it is somehow commonly accepted that we can use this degree to estimate the complexity of solving a polynomial system, even though we do not have systematic empirical data or a theory to support such a claim. In this paper, we would like to clarify the situation with the help of experiments. We first define a concept of solving degree for a polynomial system. The key question we then need to clarify is the connection of solving degree and the degree of regularity with focus on quadratic systems. To exclude the cases that do not represent the general situation, we need to define when a system is degenerate and when it is irreducible. With extensive computer experiments, we show that the two concepts, the degree of regularity and the solving degree, are related for irreducible systems in the sense that the difference between the two degrees is indeed small, less than 3. But due to the limitation of our experiments, we speculate that this may not be the case for high degree cases.

Developable surfaces have many desired properties in the manufacturing process. Since most existing CAD systems utilize tensor-product parametric surfaces including B-splines as design primitives, there is a great demand in industry to convert a general free-form parametric surface within a prescribed global error bound into developable patches. In this paper, we propose a practical and efficient solution to approximate a rectangular parametric surface with a small set of 􀀀-joint developable strips. The key contribution of the proposed algorithm is that, several optimization problems are elegantly solved in a sequence that offers a controllable global error bound on the developable surface approximation. Experimental results are presented to demonstrate the effectiveness and stability of the proposed algorithm.

Let G be a connected Lie group. We show that all characteristic classes of G are bounded—when viewed in the cohomology of the classifying space of the group G with the discrete topology—if and only if the derived group of the radical of G is simply connected in its Lie group topology. We also give equivalent conditions in terms of stable commutator length and distortion.

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We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces in$R$^{$n$}. The resulting conditions are close to optimal.