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 .00191575) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000056144) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00328601) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00523029) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00819592) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00354236) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00281477) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0298923) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000608122) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00037192) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000370598) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00259655) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00289763) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00375094) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00391632) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00246409) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00333534) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00282632) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00309355) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00324759) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012566) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000041974) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011142) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010856) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000037448) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010648) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00171297) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003508) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034962) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000357886) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000326266) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0011002) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00130142) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00021721) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0001666) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000368994) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000357368) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0014054) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00159796) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011394) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010454) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000018564) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000016992) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00825182 #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 .00195749) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000058198) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00336884) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00537316) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00831231) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00357763) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00286029) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00297795) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000577032) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000379204) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000382922) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00249184) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00292001) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00378456) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00394318) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00245542) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00336769) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00277678) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0030223) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00322238) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001127) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000359) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011332) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001031) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033874) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010202) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00161527) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003553) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034776) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000348108) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000323172) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00107695) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00127441) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00021032) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000158926) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000347604) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000342868) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0013838) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00157555) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010394) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010166) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00669324) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00611699) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000262624) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000260548) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000078638) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00007519) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011732) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010946) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0080812 #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.