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Download e-book for kindle: A treatise on algebraic plane curves by Julian Lowell Coolidge

By Julian Lowell Coolidge

ISBN-10: 0486495760

ISBN-13: 9780486495767

A radical creation to the idea of algebraic airplane curves and their family to varied fields of geometry and research. nearly totally restrained to the homes of the final curve, and mainly employs algebraic technique. Geometric tools are a lot hired, notwithstanding, specifically these related to the projective geometry of hyperspace. 1931 version. 17 illustrations.

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Example text

12. •. \In). 13. 11) contains a unique minimal generating set. That is, prove there is a subset G ç J such that J = (G) and for ail subsets F ç J with J = (F) we have G ç F. [Hint: Prove first that if J = (xn" ... 14. m as follows: (""," '''m) < (;J" ... ) Now we define an order

Qn such that Ui 2: 0 for i = l" .. ,n and 01 . U = 2:7=1 aljUj > L:7=1 CYpjUj = Qe . u for aU = 2, ... 14). We will use the following result from linear algebra (see, for example, [Ga]): THEOREM. Let A be any r x n matrix with rational entries, then exactly one of the following two alternatives holds: • there exists a row vector v E with non-negative coardinates such that the coordinates of the vector vA are ail negative or zero; • There exists a column vectar U E IQn with non-negative coordinates such that the coordinates of the vector Au are all positive.

Let f = y 2x + l, ft = yx - y, h = y2 - x E Q[x, y] with the deglex term ordering with y > x. We consider the term X = y 2 x in f. We have that f y2 + 1 = f - yft, and f ~ x 2 + 1 = f - xJz. Note that X = L = lcm(lp(ft), lp(h)) = y 2 x, and that the ambiguity introduced is _y2 + x 2 = yft - xh = S(ft, h). Also, note that S(ft, h) E (ft, h), and that it ean be reduced: S(fl, h) ~ x 2 - x. The polynomial x 2 - x is now redueed -"-+ with respect to {ft, h}, hut is not zero. Now that we have introduced S-polynomials as a way to "cancel" leading terms and to account for the ambiguity in the Division Algorithm, we can go ahead with a strategy fot computing Grübner bases.

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A treatise on algebraic plane curves by Julian Lowell Coolidge


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