New PDF release: An Introduction to Grobner Bases (Graduate Studies in

+ rI. Then, since d = c oF 0 we see that 9 ~ 9 - eX 9i ~ + Tl. + r also, we see r = Tl, as desired (by the assumption that the remainder is unique). • Case 3. d # 0 and d # c. Set h = 9 - dX 9i' Then the coefficient of X Ip(9i) in h is O. Since d # 0 we have 9 ~ h. Also, since d # c we have 9 - cXgi ~ h. So if h ~+ T2, Bueh that T2 is reduced, we get 9 ~ h ~ + T2 and 80 T2 = T, since the remainder. is unique. And 80 9 - cXgi ~ h --S+ T, as desired.

8. 1O. J = yx-x and 12 = y2 - x. J,h}. We use deglex with y> x. 10 that J ~+ 0 and J ~+ x 2 - x, the latter being reduced with respect to F. 7, F is not a Grübner basis. We can see this in another way. J,h) and J ~+ x 2 - x we have x 2 - xE (h, 12). J) = xy orlp(h) = y2. 1), Fis not a Grobner basis. 9. [x, y, z]. [x, y, z] with x < y < z. We will prove that G is a Grübner basis for J. Suppose to the contrary that there exists J E J such that lt(f) ~ (lt(91),lt(92)) = (z, y). Then, z does not divide lt(f), and y does not divide lt(f).

Let h = xy - x, 12 = x 2 - y E lQl[x, y] with the deglex term order with x < y. Let F = {h, h}. Then 8(h, 12) = xh - yh = y2 - x 2 ...!... y2 _ y, and h = y2 - Y is reduced with respect to F. Sa we add h to F, and , F' = {h, fz, h}· Then 8(h, 12) -----+ O. Now 8(h, h) = Yh - xh = 0, and F' F' 8(fz, h) = y 2h - x 2h = _y3 + x 2y -----+ x 2y - y2 -----+ O. Thus {h, fz, h} is a let F Grobner basis. 1. 8. Given F = {h, ... 1) will produce a Griibner basis Jor the ideal 1 = (h,··· ,J,). PROOF. We first need to show that this algorithm terminates.

Download PDF sample

An Introduction to Grobner Bases (Graduate Studies in Mathematics, Volume 3) by Philippe Loustaunau, William W. Adams


by Mark
4.3

Rated 4.75 of 5 – based on 23 votes