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 .00131792) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003931) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00242944) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00369228) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00585648) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00246382) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00196868) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00209171) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000422982) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000272102) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00027396) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00166812) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0019965) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00260539) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00270071) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00168357) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .002294) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00191044) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00213999) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0022866) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010312) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002529) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006574) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000677) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023838) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006622) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0011892) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023224) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024554) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00027337) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000249206) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000776298) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000929898) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00015563) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000121264) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000244808) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000238934) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00097239) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00111188) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006984) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006912) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00001264) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000011984) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00492326 #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 .00132163) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040208) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00240496) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00376814) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00586756) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00253456) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0019947) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00211554) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00042689) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000303264) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000278076) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00169597) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00208934) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0189682) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00281533) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00177783) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00235011) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0019264) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00219593) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00234901) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000975) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026028) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000836) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006864) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025638) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006914) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00124252) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002566) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040076) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00028879) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00024806) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000782298) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000920446) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000155142) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000119772) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000252472) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000241248) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00099185) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0011218) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000693) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007072) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00522056) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00460076) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000229048) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000217382) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000056876) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000068454) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009854) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008426) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00501784 #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.