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 .00114893) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036832) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00205849) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00336282) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00535594) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00234489) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00187286) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00192906) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000379524) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00025693) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000250045) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00152554) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00178359) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00235143) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00242513) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00153103) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207908) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00173431) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00192226) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0020537) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007461) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026572) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008512) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007717) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027689) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007444) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00109259) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026758) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024163) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000235205) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000220037) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00073316) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000843772) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000143358) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000113307) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000233139) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000216895) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000891843) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00102006) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007112) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000844) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000012062) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000011024) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00443453 #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 .00115855) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003788) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00205809) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00339857) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00539493) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00236336) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00189355) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00197246) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000393238) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000256991) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000256554) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00156944) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00183025) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00929417) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00217941) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00141277) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00188061) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0015456) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00171866) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00181286) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007321) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025403) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007142) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007646) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024165) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007301) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000999243) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024984) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000020952) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00021187) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000196396) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00064943) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000754291) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000126767) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000104507) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000208649) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000197186) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000798109) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00091345) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000649) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007736) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00399181) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00364325) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000190698) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000187961) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000036801) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000035638) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007503) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009221) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00426576 #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.