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 .00150707) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000614) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00373436) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00415143) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0464025) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .007601) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00448535) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00341463) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000602982) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0002967) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000300164) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00190717) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00468624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00498521) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00354058) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00221811) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00615438) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00317868) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00280771) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00337534) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016281) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000042888) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028269) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025331) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000055948) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013194) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00152856) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000057718) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034609) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000330027) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000349437) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000981) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0011239) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000173903) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000179188) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000304111) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000268872) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00120674) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00188217) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013729) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016658) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00002911) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000019909) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00777406 #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 .00153967) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000085371) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00331872) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00462256) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00900692) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00312665) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00253913) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00273131) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000661531) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000490969) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000344523) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00251723) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00231943) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00370517) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0034861) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00313035) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00360205) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00230812) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00240624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00284902) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014342) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004528) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011155) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038123) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000061019) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001559) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00144489) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000055653) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038086) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000573214) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000262947) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0015875) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00108851) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000173738) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000140845) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000301908) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000591642) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00142595) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00294717) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018335) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038252) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00582085) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00812966) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000393071) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000235532) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000058633) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00005297) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001529) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016024) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0102427 #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.