Concepts inSymbolic solution polynomial equation systems with symmetry

Algebraic equation

In mathematics, an algebraic equation, also called polynomial equation over a given field is an equation of the form where P and Q are polynomials over that field. For example is an algebraic equation over the rationals. Two equations are equivalent if they have the same set of solutions. In particular the equation is equivalent with . It follows that the study of algebraic equations is equivalent to the study of polynomials.
more from Wikipedia

Symmetry

Symmetry (from Greek ¿¿¿¿¿¿¿¿¿¿ symmetría "measure together") generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.
more from Wikipedia

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. See also: Dihedral symmetry in three dimensions.
more from Wikipedia

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are n! possible permutations of a set of n symbols, it follows that the order (the number of elements) of the symmetric group Sn is n!.
more from Wikipedia

Buchberger's algorithm

In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. It was invented by Austrian mathematician Bruno Buchberger. One can view it as a generalization of the Euclidean algorithm for univariate GCD computation and of Gaussian elimination for linear systems.
more from Wikipedia

In mathematics, more specifically in the field of group theory, a solvable group (or soluble group) is a group that can be constructed from abelian groups using extensions. That is, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation.
more from Wikipedia

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.
more from Wikipedia