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 .00131455) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040848) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00240815) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00374038) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00584007) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00249192) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00197518) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00209287) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000422562) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000272294) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000272258) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00169977) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00199181) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00260185) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00269695) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00168643) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0023177) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00191564) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0021281) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00228086) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001075) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002529) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006744) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007028) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024338) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006764) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00117968) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022942) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024532) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000272118) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000247994) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000780582) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000921732) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000154794) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000121422) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00024835) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000240712) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000993824) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00111558) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008526) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000704) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000012842) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000011892) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00491991 #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 .00132385) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039944) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00237992) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0037444) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00588187) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00253689) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00203954) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0021259) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000427884) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000274392) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000273946) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170825) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207605) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0160321) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00272608) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00177559) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00231044) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00191602) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00214956) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00228406) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010282) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025052) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006794) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008388) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025188) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006814) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00116575) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025782) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023812) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000274488) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000257966) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00077198) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000914286) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000155022) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0001201) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000247808) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000236924) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000978744) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00112188) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008064) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007116) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .004923) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00448392) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000224528) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000216846) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000052882) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00004844) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009742) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007548) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00495056 #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.