Concepts inComputation of canonical forms for ternary cubics
Cubic function
In mathematics, a cubic function is a function of the form where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function. Setting ¿(x) = 0 produces a cubic equation of the form: Usually, the coefficients a, b,c, d are real numbers. However, most of the theory is also valid if they belong to a field of characteristic other than 2 or 3.
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Canonical form
Generally, in mathematics, a canonical form (often called normal form or standard form) of an object is a standard way of presenting that object. Canonical form can also mean a differential form that is defined in a natural (canonical) way; see below. Finding a canonical form is called canonization. In some branches of computer science the term canonicalization is adopted.
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Triangular decomposition
In computer algebra, a triangular decomposition of a polynomial system is a set of simpler polynomial systems such that a point is a solution of if and only if it is a solution of one of the systems . When the purpose is to describe the solution set of in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficient of are real numbers, then the real solutions of can be obtained by a triangular decomposition into regular semi-algebraic systems.
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Moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
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Algebraic variety
This article is about algebraic varieties. For the term "variety of algebras", and an explanation of the difference between a variety of algebras and an algebraic variety, see variety (universal algebra). In mathematics, an algebraic set is the set of solutions of a system of polynomial equations. Algebraic sets are sometimes also called algebraic varieties, but normally an algebraic variety is an irreducible algebraic set, i.e. one which is not the union of two other algebraic sets.
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Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'.
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Symmetry group
In abstract algebra, the symmetry group of an object is the group of all isometries under which the object is invariant with composition as the operation. It is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in wider contexts; see below.
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