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 .00731578) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000197767) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0107035) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0183806) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0276634) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0129737) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0102672) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0101834) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00183727) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00137239) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00133121) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00911964) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0104946) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .013798) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0141337) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0092857) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0127106) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0104431) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0115859) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .012247) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039066) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011918) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003604) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003494) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000146333) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035633) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0060326) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001497) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000114207) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00101752) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000967873) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00384133) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00447137) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000752327) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000540867) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0013337) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00130956) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00530411) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00589428) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000054853) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004978) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00005828) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00005524) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0261112 #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 .00736067) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000194774) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0107113) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0183213) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0277805) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .012971) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0102363) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0102206) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00185694) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00133255) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00136223) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00910133) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0105009) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0137738) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0141637) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00930594) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0126737) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0105552) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0116646) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0122777) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004038) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000148066) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036053) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038447) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00013078) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036007) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00600463) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001444) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000119259) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00100867) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000973427) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00385846) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00449799) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000752608) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000529646) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00130569) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00131881) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00527308) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00590459) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000037394) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035914) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0250174) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0232733) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00119023) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00112688) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000286153) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00026286) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040434) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038027) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0260894 #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.