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 .00193389) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000055088) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00327368) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00528173) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00824) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0035553) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00280324) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00296744) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000567502) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000372208) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000374786) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00242532) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00291053) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00379991) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00390248) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00253845) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00338018) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0028044) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00310954) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00331028) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012794) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003503) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010312) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001017) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032586) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010146) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00166092) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033012) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038568) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000348624) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000328048) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00107407) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0012822) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000212168) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00016262) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000361124) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000363474) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00141825) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00160377) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010174) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001263) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000018554) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00001757) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00837064 #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 .0191285) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000070828) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00366001) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00529235) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00823402) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0036141) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00285628) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00298608) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000579588) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00037585) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000374666) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0024974) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00298504) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00381657) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00400666) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0024802) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00346947) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00282175) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00310931) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0033227) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011994) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035086) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001123) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001021) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003491) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011442) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00171668) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035182) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034882) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000351426) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000327426) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00108394) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00130361) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000216672) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000164586) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000374354) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000359248) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00144222) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00160466) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001069) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010258) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00721497) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00643773) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000272378) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000272402) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00008409) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000081958) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012584) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010922) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00848998 #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.