# Dimension of an algebraic variety

I would like to compute, using Groebner bases, the dimension of the variety defined by a set of polynomials in several variables. In the wikipedia page the method is described, and an implementation is possible, but I was hoping there was a predefined function that would do that.

• Is this about math or Mathematica? Commented Nov 14, 2013 at 12:52
• Dear Yves, this was a question about Mathematica, I wanted to know if there is any predefined function that computes the dimension by implementing the algorithm cited above. Commented Nov 15, 2013 at 17:05
• Thank you for the info - in that case you are in good hands with Daniel :) Commented Nov 16, 2013 at 6:45

I show some aged code cribbed from this MathGroup post (See also this library item). There is some amount of explanation in each. But the most salient quote, from the post, is this. "It is unlikely that I can do justice to the explanation. So far as correctness goes, let's just say the code is on the honor system."

firstContainsSecond[l1_, l2_] := (Union[l1, l2] === l1)

isIndependentSet[set_, sets_] :=
Map[! firstContainsSecond[set, #] &, sets]

getMaxIndependentSets[vars_, inset_, heds_, maxlen_, indx_, sets_] :=
Module[{currentset, vlen = Length[vars], ilen = Length[inset],
enlarged = False, newmax = maxlen, maxsets = sets},
Do[If[ilen + vlen - i + 1 <= maxlen, Break[]];
currentset = Append[inset, vars[[i]]];
If[isIndependentSet[currentset,
heds], {maxsets, enlarged, newmax} =
getMaxIndependentSets[vars, currentset, heds, newmax, i + 1,
maxsets];
If[! enlarged, maxsets = {maxsets, currentset};
newmax = Max[newmax, Length[currentset]];
enlarged = True;];], {i, indx, vlen}];
{maxsets, enlarged, newmax}]

idealDimension[ideal_, vars_] :=
Module[{ord = MonomialOrder -> DegreeReverseLexicographic, gb,
gb = GroebnerBasis[ideal, vars, ord];
Map[First[First[#]] &,
First[InternalDistributedTermsList[gb, vars, ord]]];
heds = Apply[
And, (Map[(vars*#) &, (pheads /. _?Positive -> 1)]/.0 ->
Sequence[])];
{maxsets, el, mlen} =
getMaxIndependentSets[vars, {}, heds, 0, 1, {}];
mlen]


This comes with no warranties. If ever there were any, they must have expired by now.

• Thanks a lot Daniel! I will try the code and let you know how it worked ;) Commented Nov 15, 2013 at 10:25
• Daniel, I managed to get the code working in Mathematica 9, I will post it here in a couple of days after some debugging. Thank you very much for your help! Commented Nov 16, 2013 at 8:27

Here's the code kindly provided by Daniel Lichtblau in the accepted answer, modified to work with Mathematica 9. All the credit goes to him, and, of course, this comes with no warranties ;) The code is inspired from T.Becker, V.Weisphenning - Groebner bases, (Table 9.6, pp 449).

firstContainsSecond[l1_, l2_] := (Union[l1, l2] === l1);
isIndependentSet[set_, sets_] :=
Map[! firstContainsSecond[set, #] &, sets];
getMaxIndependentSets[vars_, inset_, heds_, maxlen_, indx_, sets_] :=
Module[{currentset, vlen = Length[vars], ilen = Length[inset],
enlarged = False, newmax = maxlen, maxsets = sets},
Do[
If[ilen + vlen - i + 1 <= maxlen, Break[]];
currentset = Append[inset, vars[[i]]];
If[And @@
isIndependentSet[currentset, heds], {maxsets, enlarged,
newmax} =
getMaxIndependentSets[vars, currentset, heds, newmax, i + 1,
maxsets];
If[! enlarged, maxsets = Append[maxsets, currentset];
newmax = Max[newmax, Length[currentset]];
enlarged = True;];], {i, indx, vlen}];
{maxsets, enlarged, newmax}
];
idealDimension[ideal_, vars_] :=
Module[{ord = DegreeReverseLexicographic, gb, pheads, heds, maxsets,
el, mlen},
gb = GroebnerBasis[ideal, vars, MonomialOrder -> ord];
pheads = Map[First[MonomialList[#, vars, ord]] &, gb];

• Thanks. I had forgotten about the context change to DistributedTermsList. And the "new" MonomialList did not exist when I wrote that. Ironically, version 3 also had a MonomialList that was related to the current one. I no longer recall what internal misgivings caused it to go away, only to reappear a few versions later. Commented Nov 16, 2013 at 23:16
• Why does the result depend on the order of variables? e. g. idealDimension[{-1 + a^2 + b^2 + d^2 + e^2, b d - a e + c e - b f, a b + b c + d e + e f, -1 + b^2 + c^2 + e^2 + f^2} , {a, b, c, d, e, f}] gives 3 but with the reversed order {f, e, d, c, b, a}` the result is 4.