Concepts inPartial degree formulae for rational algebraic surfaces
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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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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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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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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Parametric equation
the butterfly curve. ]] In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion. Abstractly, a parametric equation defines a relation as a set of equations. Therefore, it is somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.
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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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Degree of a polynomial
The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e. as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The word degree is now standard, but in some older books, the word order may be used instead. For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .
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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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