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 .0013437) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000039312) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00242454) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00377739) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00592394) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .0025245) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00200943) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00212422) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000432182) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00027869) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000278704) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0017065) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00203632) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00266074) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0027711) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00174541) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00235164) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00195909) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00217073) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00233739) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000008746) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00002661) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000008168) #primes = 2 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00000703) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000026386) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007152) #primes = 4 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00118987) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00002457) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000024332) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000266186) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000250248) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000787548) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000952396) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000157976) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000123132) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000266414) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .0002481) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000993914) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .00114595) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007198) #primes = 7 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00000718) #primes = 8 #prunedViaCodim = 0
Strategy: IndependentSet (time .000011364) #primes = 9 #prunedViaCodim = 0
Strategy: IndependentSet (time .000012214) #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00505034
#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 .00135474) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00003913) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00239959) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00382463) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00595045) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00256714) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00204568) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00215448) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00043566) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .00028366) #primes = 0 #prunedViaCodim = 0
Strategy: Factorization (time .000281702) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00174422) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00211542) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00270455) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00280792) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .0156844) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00240256) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00199455) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00221676) #primes = 0 #prunedViaCodim = 0
Strategy: Linear (time .00232858) #primes = 0 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000008676) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .0000255) #primes = 1 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006924) #primes = 2 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007314) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000024618) #primes = 3 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000006842) #primes = 4 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00119883) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000026638) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00002389) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000273082) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000248944) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000798124) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .00092486) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000158794) #primes = 6 #prunedViaCodim = 0
Strategy: Factorization (time .000123912) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000262866) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .000243308) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .00101822) #primes = 6 #prunedViaCodim = 0
Strategy: Linear (time .00113009) #primes = 6 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000007128) #primes = 7 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000008192) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .00502853) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .00444162) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .000219808) #primes = 8 #prunedViaCodim = 0
Strategy: Birational (time .000219154) #primes = 8 #prunedViaCodim = 0
Strategy: Linear (time .000052548) #primes = 8 #prunedViaCodim = 0
Strategy: Linear (time .000087126) #primes = 8 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .00000968) #primes = 9 #prunedViaCodim = 0
Strategy: DecomposeMonomials(time .000008518) #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00502786
#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.