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The direct approach can be made, by means of the function field of a variety (i.e. rational functions): take the G-invariant rational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put it in the ...
In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties: such that [1] (i) The map π {\displaystyle \pi } is surjective, and its fibers are exactly the G-orbits in X.
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and () The quotient rule states that the derivative of h(x) is
The book has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication. Marvin Minsky said in his paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems." [27]
The question is of a great interest since the GIT approach produces an explicit quotient, as opposed to an abstract quotient, which is hard to compute. One known partial answer to this question is the following: [ 2 ] let X {\displaystyle X} be a locally factorial algebraic variety (for example, a smooth variety) with an action of G ...
If k(X) is the field of rational functions on X, then any non-zero f ∈ k(X) may be written as a quotient g / h, where g and h are in ,, and the order of vanishing of f is defined to be ord Z (g) − ord Z (h). [4] With this definition, the order of vanishing is a function ord Z : k(X) × → Z.