Rationalisoida

Rationalisoida is a theoretical procedure in algebraic geometry and computational mathematics that refers to transforming a given object, such as a curve, surface, or function, into a rational representation. The goal is to obtain a parametrization or map expressed entirely by rational functions in one or more parameters, while preserving the essential properties of the original object under a specified equivalence (for example birational equivalence or functional equivalence over a chosen field).

Origin and terminology: The term is a relatively recent coinage used in contemporary discussions about symbolic computation and rational approximation. It combines the idea of rational expressions with a notion of re-expression or reparametrization. The concept appears in various forms across mathematical literature, often in the context of studying when and how objects can be modeled by rational maps.

Methods and theory: In practice, rationalisoida involves either (1) constructing a rational parametrization by eliminating radicals or transcendental components through algebraic manipulation and elimination theory, or (2) producing a rational surrogate that matches the original object's invariants within a defined tolerance. The method is often facilitated by tools such as Gröbner bases, resultants, and other techniques from elimination theory, and may rely on birational maps for curves and surfaces.

Applications: The procedure is used in algebraic geometry to study curves and surfaces, in computer-aided geometric design to enable stable rational representations, and in symbolic computation to simplify expressions or enable exact arithmetic with rational functions. It also informs numerical methods where rational parametrizations improve stability and reproducibility.

Limitations: Rationalisoida is not always possible; some objects do not admit a global rational parametrization. Local parametrizations may introduce singularities or require piecewise definitions. The process can also increase algebraic complexity and may introduce approximation errors when surrogate models are employed.

See also: Birational mapping, Rational parametrization, Isomorphism, Elimination theory, Gröbner basis.