Hi Andrew, On 12 Okt., 15:19, andrew ewart <aewartma...@googlemail.com> wrote: > i think its just reffering to vector space dimension > I have no idea what the Krull dimension of this space is > Also if i try lex in QQ the grobner basis i get out is > [x + y + z - 3, y^2 + y*z - 3*y + z^2 - 3*z + 2, z^3 - 3*z^2 + 2*z + > 2/3]
My guess about your notions is as follows: By "size", you mean number of points. Hence, in order to make sense, by "size of a variety", you mean "number of points of a zero- dimensional variety". Let V be the variety and I the ideal (contained in a polynomial ring P over a field K) defining V. Lets assume further that V is defined over the algebraic closure of K (i.e., all coefficients of the defining polynomials are in K, but the points of V may live in a bigger field). If this is the case, then let's summarise some results from commutative algebra (hope I remember everything correctly, I have no textbook handy...): 1. V is zero-dimensional if and only if I is of Krull dimension zero. 2. The dimension of P/I as a vector space over K is an upper bound for the number of points of V. 3. If I is radical then the number of points of V is equal to the vector space dimension of P/I (which is often simply referred to as the vector space dimension of I). 4. In order to compute the Krull or vector space dimension, one needs a Groebner basis. But it is not needed to use a lex order. In many cases, degrevlex is easier. Hence, you could proceed as follows (of course, you could put the computations into one line; printing intermediate results is just illustration). sage: P.<x,y,z> = QQ[] sage: P.term_order() Degree reverse lexicographic term order sage: I = Ideal(x+y+z-3,x^2+y^2+z^2-5,x^3+y^3+z^3-7) sage: IR = I.radical().groebner_basis()*P; IR Ideal (z^3 - 3*z^2 + 2*z + 2/3, y^2 + y*z + z^2 - 3*y - 3*z + 2, x + y + z - 3) of Multivariate Polynomial Ring in x, y, z over Rational Field sage: IR.dimension() 0 sage: IR.vector_space_dimension() 6 Hence, over CC (the algebraic closure of QQ), the variety V defined by contains precisely 6 points. Cheers, Simon -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
