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 .0012783) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040174) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00229866) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00355522) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00557342) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00237626) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00188484) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00198938) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00040198) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000260146) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000256816) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00161639) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00193224) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00252046) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0025917) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00162345) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00222804) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00185259) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00206104) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00217006) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010828) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002201) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006132) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007294) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022628) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007122) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0011764) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022788) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024216) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000275744) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000234582) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00073849) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000866792) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000155398) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000112764) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000238776) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000225216) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00093057) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00106628) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006522) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006224) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010016) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000009304) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00469175 #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 .00128675) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000371) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00230211) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00355153) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00559135) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00239809) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00190099) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00200474) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00040927) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000260986) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000260504) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0016651) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00196371) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .017435) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00260171) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00167222) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00221408) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00185094) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0020726) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0022019) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009296) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023832) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005976) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006938) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023236) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006002) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .001166) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022618) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022114) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000263866) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000235104) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000745258) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000870788) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00014508) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000113132) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00024096) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000229218) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00095109) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00106873) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006824) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000721) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00479681) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00428555) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00020143) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000214058) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000053596) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000048374) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009532) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006996) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00478281 #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.