Given an ideal in a polynomial ring, or a quotient of a polynomial ring whose base ring is either QQ or ZZ/p, return a list of minimal primes of the ideal.
i1 : R = ZZ/32003[a..e] o1 = R o1 : PolynomialRing |
i2 : I = ideal"a2b-c3,abd-c2e,ade-ce2"
2 3 2 2
o2 = ideal (a b - c , a*b*d - c e, a*d*e - c*e )
o2 : Ideal of R
|
i3 : C = minprimes I; |
i4 : netList C
+---------------------------+
o4 = |ideal (c, a) |
+---------------------------+
| 2 3 |
|ideal (e, d, a b - c ) |
+---------------------------+
|ideal (e, c, b) |
+---------------------------+
|ideal (d, c, b) |
+---------------------------+
|ideal (d - e, b - c, a - c)|
+---------------------------+
|ideal (d + e, b - c, a + c)|
+---------------------------+
|
i5 : C2 = minprimes(I, Strategy=>"NoBirational", Verbosity=>2)
Strategy: Linear (time .00123911) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000037173) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00202311) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00304012) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00468038) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00210589) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00164115) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00171387) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000317389) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000225287) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000220526) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0014485) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0016793) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00216769) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00250414) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00144421) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0019571) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00167356) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00180342) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00190987) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007118) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000031258) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005943) #primes = 2 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006698) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00002209) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005944) #primes = 4 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00147165) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000024187) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000022074) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000204617) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000177952) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000644416) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000736386) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000120067) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000092227) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000216682) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .00019837) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .00167483) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000981017) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007364) #primes = 7 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007127) #primes = 8 #prunedViaCodim = 0
Strategy: IndependentSet (time .000013216) #primes = 9 #prunedViaCodim = 0
Strategy: IndependentSet (time .000009941) #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00404344
#minprimes=6 #computed=10
2 3
o5 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b),
------------------------------------------------------------------------
ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)}
o5 : List
|
i6 : C1 = minprimes(I, Strategy=>"Birational", Verbosity=>2)
Strategy: Linear (time .00111929) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000033156) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00183412) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .013291) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .01527) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00222561) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00750026) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00175575) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000333452) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000226121) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000219692) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .16209) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00176942) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00257944) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00227993) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0309309) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0256724) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00165587) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00182335) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00193925) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007277) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000022536) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006761) #primes = 2 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005945) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000021124) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007129) #primes = 4 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00103794) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000021829) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000020751) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000224822) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .00018034) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000658442) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000751781) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000119764) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000092193) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .0002205) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .00020982) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000854115) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000963813) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005902) #primes = 7 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00000618) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .00638694) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .00401841) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .000194187) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .000284562) #primes = 8 #prunedViaCodim = 0
Strategy: Linear (time .00004788) #primes = 8 #prunedViaCodim = 0
Strategy: Linear (time .000044331) #primes = 8 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000009491) #primes = 9 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000013064) #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00437682
#minprimes=6 #computed=10
2 3
o6 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b),
------------------------------------------------------------------------
ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)}
o6 : List
|
This will eventually be made to work over GF(q), and over other fields too.