But, as immediately below, in some cases we must consider parameter values in the algebraic closure of the subfield. To simplify notation we generally work with coefficients in the real numbers, but it should be understood that one could work in any subfield of the complex numbers as well. Where and the common denominator are all polynomials of degree or less written in descending degree. This lets us rephrase Theorem B in another useful form, where the can be found directly from the expression of a rational function in the form That is, every rational curve is a transform of a normal curve. In Section 5 we observe that the rational normal curve in or, , is universal for rational curves. Theorem B does clarify that, while the degree bounds the size of a linear set, the curve may lie in a smaller dimensional linear set. The generalization is to rational curves and we can give the dimensions of the smallest linear space containing the curve. We then generalize and rephrase our result in Section 4 as Theorem B. Instead we can view these results from an affine point of view using the built-in function, which we discuss in Section 3. Unfortunately, projective geometry is not computationally friendly. We give the linear algebra proof in Section 2. He also considers the degree versus dimension issue in a number of other situations. This theorem, as well as many of the other facts in this article, is given in Joe Harris ’s book from a projective geometry point of view. Then for any, the curve lies in a linear subset of or of dimension. Let, be a curve in or where the coordinate functions are polynomials of degree or less.
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