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 .0013845) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039474) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0024144) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0037916) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00596025) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00254814) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00202322) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0021661) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00043232) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00027882) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000278314) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170949) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00206705) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00270396) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0027793) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00173993) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00237983) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00196798) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00220786) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00233195) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007718) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002615) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000979) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007142) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025964) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006858) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00119545) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025012) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025282) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000268544) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0002513) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00079342) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000936774) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000157528) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000123264) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000252544) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00024788) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00101563) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00114721) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000693) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007166) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000011288) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000125) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00502669 #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 .00136544) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000037954) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00244306) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0037997) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00602264) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00255617) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .002063) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00216554) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000450204) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000285634) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000282592) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00173637) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00210894) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00272991) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00283471) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0139522) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00244396) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00204522) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00219956) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00233392) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008442) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025766) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006998) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007148) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025202) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007014) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00122926) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027298) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025574) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000274502) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00025) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0007858) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00092994) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00015841) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00012332) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000253542) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00025718) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00100453) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00114002) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007958) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007264) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0049815) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00455381) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00021902) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000217222) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000051586) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00005026) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008402) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007804) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00514336 #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.