The routine reduces the target of M by elementary moves (see elementary) involving just d+1 variables. The outcome is probabilistic, but if the routine fails, it gives an error message.
See the book of Evans and Griffith (Syzygies. London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985.)
i1 : kk=ZZ/32003
o1 = kk
o1 : QuotientRing
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i2 : S=kk[a..e]
o2 = S
o2 : PolynomialRing
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i3 : i=ideal(a^2,b^3,c^4, d^5)
2 3 4 5
o3 = ideal (a , b , c , d )
o3 : Ideal of S
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i4 : F=res i
1 4 6 4 1
o4 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o4 : ChainComplex
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i5 : f=F.dd_3
o5 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | a2 0 0 d5 |
{7} | 0 -b3 -c4 0 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6 4
o5 : Matrix S <-- S
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i6 : EG = evansGriffith(f,2) -- notice that we have a matrix with one less row, as described in elementary, and the target module rank is one less.
o6 = {5} | c4 d5 0
{6} | -b3 0 d5
{7} | 0 -b3 -8251a4+5071a3b-9480a2b2+12365a3c+8231a2bc+5026a2c2-c4
{7} | a2 0 2653a4-6203a3b-11950a2b2-13508a3c+5864a2bc+10259a2c2
{8} | 0 a2 -7501a3+9534a2b-7216a2c
------------------------------------------------------------------------
0 |
0 |
-8251a2b3+5071ab4-9480b5+12365ab3c+8231b4c+5026b3c2 |
2653a2b3-6203ab4-11950b5-13508ab3c+5864b4c+10259b3c2+d5 |
-7501ab3+9534b4-7216b3c-c4 |
5 4
o6 : Matrix S <-- S
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i7 : isSyzygy(coker EG,2)
o7 = true
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