Concepts inPartial degree formulae for rational algebraic surfaces
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of dimension two).
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Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.
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Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language. In science, a specific formula is a concise way of expressing information symbolically as in a mathematical or chemical formula. The plural of formula can be spelled either formulae (like the original Latin) for mathematical or scientific senses, or formulas for more general senses.
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Rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
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Rational surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques-Kodaira classification of complex surfaces, and were the first surfaces to be investigated.
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Resultant
In mathematics, the resultant of two univariate polynomials and is a polynomial function of their coefficients that is zero if and and only if the two polynomials have a common root in an algebraically closed field containing the coefficients. Alternatively the resultant is sometimes defined for two homogeneous polynomials in two variables, in which case it vanishes when the polynomials have a common non-zero solution, or equivalently when they have a common zero on the projective line.
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Parametrization
Parametrization (or parameterization; also parameterisation, parametrisation in British English) is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object. Sometimes, this may only involve identifying certain parameters or variables.
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Implicit and explicit functions
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R(x, y) = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval. Geometrically, the graph defined by R(x,y) = 0 will overlap locally with the graph of an equation y = f(x).
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