A.1.1 Basic programming

We show in the example below the following:

• define the ring R of characteristic 32003, variables x,y,z, monomial ordering dp (implementing F_32003[x,y,z])
• list the information about R by typing its name
• check the order of the variables
• define the integers a,b,c,t
• define a polynomial f (depending on a,b,c,t) and display it
• define the jacobian ideal i of f
• compute a Groebner basis of i
• compute the dimension of the algebraic set defined by i (requires the computation of a Groebner basis)
• create and display a string in order to comment the result (text between quotes " "; is a 'string')
• load a library (see primdec_lib)
• compute a primary decomposition for i and assign the result to a list L (which is a list of lists of ideals)
• display the number of primary components and the first primary and prime components (entries of the list L[1])
• implement the localization of F_32003[x,y,z] at the homogeneous maximal ideal (generated by x,y,z) by defining a ring with local monomial ordering (ds in place of dp)
• map i to this ring (see imap) - we may use the same name i, since ideals are ring dependent data
• compute the local dimension of the algebraic set defined by i at the origin (= dimension of the ideal generated by i in the localization)
• compute the local dimension of the algebraic set defined by i at the point (-2000,-6961,-7944) (by applying a linear coordinate transformation)

ring R = 32003,(x,y,z),dp; R; ==> // coefficients: ZZ/32003 ==> // number of vars : 3 ==> // block 1 : ordering dp ==> // : names x y z ==> // block 2 : ordering C x > y; ==> 1 y > z; ==> 1 int a,b,c,t = 1,2,-1,4; poly f = a*x3+b*xy3-c*xz3+t*xy2z2; f; ==> 4xy2z2+2xy3+xz3+x3 ideal i = jacob(f); // Jacobian Ideal of f ideal si = std(i); // compute Groebner basis int dimi = dim(si); string s = "The dimension of V(i) is "+string(dimi)+"."; s; ==> The dimension of V(i) is 1. LIB "primdec.lib"; // load library primdec.lib list L = primdecGTZ(i); size(L); // number of prime components ==> 6 L[1][1]; // first primary component ==> _[1]=2y2z2+y3-16001z3 ==> _[2]=x L[1][2]; // corresponding prime component ==> _[1]=2y2z2+y3-16001z3 ==> _[2]=x ring Rloc = 32003,(x,y,z),ds; // ds = local monomial ordering ideal i = imap(R,i); dim(std(i)); ==> 1 map phi = R, x-2000, y-6961, z-7944; dim(std(phi(i))); ==> 0