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 .00308366) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000096068) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00518354) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00842258) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0131961) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00582577) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00466562) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0492834) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00092851) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00061256) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000604344) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00401204) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00481899) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00630109) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00647515) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00415254) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0056556) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00457196) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00512961) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00533779) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000204) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00006347) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018106) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019362) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000059656) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019252) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00273217) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000062256) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000062208) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000534768) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000507904) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00177824) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00212754) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000346298) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000266704) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0005939) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00056683) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00229541) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00274324) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019812) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019084) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000032812) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000030904) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0135455 #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 .00311022) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000099138) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00519617) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00869196) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0133767) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00595231) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00468223) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00477198) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000893336) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000618868) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000602466) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00399347) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00476494) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00620901) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00639509) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00416202) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00573624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00459145) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00505101) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00521414) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000020492) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000063038) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019132) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019404) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000060702) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018836) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00274224) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000063212) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000058688) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00054675) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00050343) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00173086) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00209183) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000335856) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000258402) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000576954) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000563794) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00226115) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00258659) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019536) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000021324) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0110169) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00996639) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000444266) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000441472) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00014604) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000133966) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023224) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000020152) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0132508 #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.