some SINGULAR language suggestions

[Bradford Hovinen (hovinen@…]

Here are a couple of language-suggestions.

Indexed variables =================

It should be possible to create a ring whose variables are indexed with any arbitrary immutable type. Furthermore, it should be possible create an indexed set of variables where the index-set is any immutable type. Suggested syntax is below.

The following creates a ring with anonymous variables, i.e. variables must be accessed through the member 'vars' of the ring-object.

L = [(0,1,3)] List (or any iterable object) with indices of ring-variables R = ring (0,L) R.vars[(0,1,3)] Variables indexed by a tuple R.vars[0,1,3] Syntactic sugar -- same meaning as [(0,1,3)]

The following creates named variables from the variables of a ring: x = R.vars x[0,1,3] Same as R.vars[0,1,3]

One could also assign specific variables x = R.vars[(0,1,3)] x Same as R.vars[0,1,3]

The following is syntactic sugar for the declaration above. It creates a ring with named variables at the same time L = [0, 1, 2, 3] Now variables are indexed by a single integer R = ring (0,(x[L], y[L])) x[0] Both ring-elements y[2] R.vars[x][0] Same as x[0]

The usual syntax ring (0, (x,y,z)) should also work R = ring (0, (x,y,z)) x All ring-elements y z R.vars[x] Same as x

The syntax R.vars[...] will be extremely useful when writing packages whose procedures work with multiple rings simultaneously. For example:

R = ring (0, (x,y,z)) x Element of R S = ring (0, (x,y)) x Element of S R.vars[x] Element x of R

Strictly speaking, the syntax should be R.varsx?, but that is a bit cumbersome and it should be possible to make the syntax R.vars[x] work.

Grading =======

It should be possible to specify degrees of generators not just with a weight-vector but with a function which assigns a degree to each index in the index-set.

def degree (I):

L = (0, 3, 4) return I[0] - L[I[1] - 1]

Cartesian cross-product of iterables L1 and L2 def product (L1, L2):

for x in L1:

for y in L2:

yield (x, y)

idx = product ([0, 3, 4] + range (6, 14), range (1, 4))

R = ring (0, x[idx], wp(degree)) Degrees given by function degree

It should be possible to have a degree-function which returns multidegrees as well.