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 .000800537) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000023938) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00135134) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00231958) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .0246024) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00173894) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00137531) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00137322) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000294125) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000220811) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00021106) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00108117) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00116883) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00157609) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00163854) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00101217) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00140509) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00116946) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00131672) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00143519) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006624) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000017446) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005293) #primes = 2 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005174) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000018571) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005865) #primes = 4 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000782418) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .0000178) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00001633) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .00015617) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000143178) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000517341) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000622206) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000107951) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000089569) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000151335) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000144704) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000621156) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000742448) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005649) #primes = 7 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005195) #primes = 8 #prunedViaCodim = 0
Strategy: IndependentSet (time .00000863) #primes = 9 #prunedViaCodim = 0
Strategy: IndependentSet (time .000007313) #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00347586
#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 .000786498) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000024954) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00141937) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00246026) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00387801) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00174306) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00145702) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00142031) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000295295) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000221748) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000217581) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00109288) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00130443) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00158543) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00164428) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00101727) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00141123) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00117137) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00132444) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00142708) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006993) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000018102) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000004827) #primes = 2 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005771) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000016319) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000004947) #primes = 4 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000761542) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000017304) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000017495) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000158967) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000147407) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000527767) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000625343) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000109962) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000092307) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000151628) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000144571) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000633547) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000733959) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006263) #primes = 7 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005696) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .00295931) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .0026918) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .000118058) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .000114176) #primes = 8 #prunedViaCodim = 0
Strategy: Linear (time .000028227) #primes = 8 #prunedViaCodim = 0
Strategy: Linear (time .000027107) #primes = 8 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006176) #primes = 9 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000005988) #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00345043
#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.