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 .0011661) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000374) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00209769) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00409157) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00300625) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00196633) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00201608) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000402576) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000297719) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00029257) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00162247) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00153909) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215352) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00221822) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00181057) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00198199) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0139991) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001046) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027602) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008055) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008696) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00113706) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027141) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022863) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000251131) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000785225) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000888628) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000174548) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000152496) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000232937) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000947149) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00109218) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007163) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008716) #primes = 7 #prunedViaCodim = 0 Strategy: IndependentSet (time .000012513) #primes = 8 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00413803 #minprimes=6 #computed=8 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 .00123417) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040406) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00211125) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00418769) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00306291) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00207571) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00204073) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000400211) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000302097) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000320402) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00162497) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00160119) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00217249) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00235961) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00183337) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207996) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00218377) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009688) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027752) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006392) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009718) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00118284) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029696) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022923) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000255941) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000800573) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000924867) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000179246) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000151033) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000246292) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00102354) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00114665) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007875) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009177) #primes = 7 #prunedViaCodim = 0 Strategy: Birational (time .00456166) #primes = 7 #prunedViaCodim = 0 Strategy: Birational (time .00018665) #primes = 7 #prunedViaCodim = 0 Strategy: Linear (time .000048771) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009408) #primes = 8 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00434218 #minprimes=6 #computed=8 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.