基于签名的Groebner基算法及其应用

科学研究和实际工程中产生的许多问题都可以转化为多项式方程组求解问题，求解多项式方程组是解决这类问题的关键。计算多项式系统的Groebner基是求解多项式方程组的有效方法。该项目将研究多项式系统签名Groebner基的相关理论和算法，给出统一形式的签名Groebner基算法（包括数域上的多项式环，可解多项代数以及局部环）， 并在计算机代数系统中加以实现。进一步， Groebner基将被应用于代数方程组求解和几何定理机器证明。

Many problems, which are appeared in scientific research and practical engineering, can be transfered into polynomial equations, and the key to solve these problems is solving these equations. Computing the Groebner bases of polynomial systems is an effective method to solve polynomail euqations. This research project will study the theories and algorithms of signature-based Groebner basis. Signature-based Groebner basis algorithm with uniforms for polynomial ring over field, solvable polynomial algebra and local ring will be proposed, and efficient implementation will be given on computer algebra systems. Further more, Groebner basis will be applied to polynomial equations solving and machine proving of geometric theorems.