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 .00130632) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000044051) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00246114) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00415575) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00620632) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00278763) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00227985) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00229994) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000430413) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000330674) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00027706) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00193307) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00211645) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00277901) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00286846) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00179186) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0024923) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00199624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00233972) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00268616) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010861) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000049366) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008951) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009769) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000045598) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002256) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00135102) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000030379) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000047403) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000298007) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000222105) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000855351) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00098002) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000156102) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000171585) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000241284) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000233142) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00102124) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00113974) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000906) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010317) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000019229) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000014391) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00518078 #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 .00127027) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000047096) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00243745) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00398942) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00665377) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .003002) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00227055) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00247792) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000464564) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000337157) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000290357) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00186061) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215782) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00295837) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00316107) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00223772) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00259912) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00213155) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00231999) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227445) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018532) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036173) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006969) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009699) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029611) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007995) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00133792) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039753) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000031632) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000277968) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000239765) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000857302) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000967595) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000200301) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000149992) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00030052) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000227677) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00109052) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00130943) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008118) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010275) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00582127) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00545789) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000288326) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000261796) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000056046) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000050406) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001028) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010869) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00540079 #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.