- Dimension of an algebraic variety
- P(F,G) = 0.
This implies that F and G are constrained to take related values (up to some finite freedom of choice): they cannot be truly independent.
For the function field even to be defined, V here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the field of fractions of the coordinate ring of V. Using polynomial equations, it is easy to define sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible.
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