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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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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Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and its applications. A "constant" in this context should not be confused with a mathematical constant which is a specific number independent of the scope of the given problem.
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Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
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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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Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x − x/4 + 7 is a polynomial, but x − 4/x + 7x is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not an integer (3/2).
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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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