math.AC

arXiv:2501.06217v1 Announce Type: cross Abstract: Kinematics of rigid bodies can be analyzed in many different ways. The advantage of using Euler parameters is that the resulting equations are polynomials and hence computational algebra, in particular Gr\"obner bases, can be used to study them. The disadvantage of the Gr\"obner basis methods is that the computational complexity grows quite fast in the worst case in the number of variables and the degree of polynomials. In the present article we show how to simplify computations when the mechanism contains revolute joints. The idea is based on the fact that the ideal representing the constraints of the revolute joint is not prime. Choosing the appropriate prime component reduces significantly the computational cost. We illustrate the method by applying it to the well known Bennett's and Bricard's mechanisms, but it can be applied to any mechanism which has revolute joints.

arXiv:2101.03482v3 Announce Type: replace-cross Abstract: We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The Gr\"obner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in $x_1$. The proper basis is defined and computed in the variables $\tilde{\bm{x}}:=(x_2,\dotsc,x_n)$ with $x_1$ serving as a parameter in the algebra $K[x_1][\tilde{\bm{x}}]$. Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of $\tilde{\bm{x}}$ unnecessarily involves the least variable $x_1$ but also M\"oller's algorithm due to its polynomial division mechanism. This is corroborated by a series of benchmark testings herein. The proper basis is in a modular form and neater than Gr\"obner basis and hence reduces its coefficient swell problem. It is expected that all the state of the art algorithms improving Buchberger's algorithm over the last decades can be further improved if we apply them to the proper basis.