A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization

In symbolic computing, a major bottleneck is middle expression swell. Symbolic geometric computing based on invariant algebras can alleviate this difficulty. For example, the size of projective geometric computing based on bracket algebra can often be restrained to two terms, using final polynomials, area method, Cayley expansion, etc. This is the "binomial" feature of projective geometric computing in the language of bracket algebra. In this paper we report a stunning discovery in Euclidean geometric computing: the term preservation phenomenon. Input an expression in the language of Null Bracket Algebra (NBA), by the recipe we are to propose in this paper, the computing procedure can often be controlled to within the same number of terms as the input, through to the end. In particular, the conclusions of most Euclidean geometric theorems can be expressed by monomials in NBA, and the expression size in the proving procedure can often be controlled to within one term! Euclidean geometric computing can now be announced as having a "monomial" feature in the language of NBA. The recipe is composed of three parts: use long geometric product to represent and compute multiplicatively, use "BREEFS" to control the expression size locally, and use Clifford factorization for term reduction and transition from algebra to geometry. By the time this paper is being written, the recipe has been tested by 70+ examples from \cite{chou}, among which 30+ have monomial proofs. Among those outside the scope, the famous Miquel's five-circle theorem \cite{chou2}, whose analytic proof is straightforward but very difficult symbolic computing, is discovered to have a 3-termed elegant proof with the recipe

Geometric algebra techniques in flux compactifications

We study `constrained generalized Killing (s)pinors', which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As an application, we show how our approach can be used to efficiently recover results pertaining to N=1 compactifications of M-theory on eight-manifolds.Comment: 70 page

Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6

We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of \Prj^1 for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.Comment: v3: full version of the pape