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 .00121798) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000057174) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00240164) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00357718) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0168849) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00403173) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00308897) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00319882) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000610924) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000417806) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000385123) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00288703) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00310193) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00389982) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00410801) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00261214) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00356471) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00296896) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00364406) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00354242) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000015994) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000053659) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012605) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016092) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000060819) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000015973) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00200405) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000051718) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000041943) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000380523) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000319308) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00118435) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00139872) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000221465) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000200886) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000407521) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000358419) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00153315) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00175344) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016184) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017859) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000021496) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000022584) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00876831 #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 .00207703) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000071071) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0035676) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00561516) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00834846) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0039208) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00313621) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .003187) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000604455) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000402744) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000440646) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00272038) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .002965) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00425158) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0039878) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00261783) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00347606) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00292693) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00307361) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00201794) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009644) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032587) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007388) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008988) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032382) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006705) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00108392) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035337) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025632) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000233187) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000170412) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000733173) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000807331) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000123391) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000109405) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000230168) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000197092) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00087937) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00100463) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008751) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008549) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00444725) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00412086) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000211618) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000165081) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000043292) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000047723) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010806) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011483) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00487549 #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.