Exercise 1 Day 1 case fair-loaded KK=QQ R=KK[p_1..p_21,x_1..x_5,y_1..y_5] J2 = ideal(p_1-x_1/6, p_2-x_1/6-x_2/6, p_3-x_1/6-x_3/6, p_4-x_1/6-x_4/6, p_5-x_1/6-x_5/6, p_6-x_1/6-(1-x_1-x_2-x_3-x_4-x_5)/6, p_7-x_2/6, p_8-x_2/6-x_3/6, p_9-x_2/6-x_4/6, p_10-x_2/6-x_5/6, p_11-x_2/6-(1-x_1-x_2-x_3-x_4-x_5)/6, p_12-x_3/6, p_13-x_3/6-x_4/6, p_14-x_3/6-x_5/6, p_15-x_3/6-(1-x_1-x_2-x_3-x_4-x_5)/6, p_16-x_4/6, p_17-x_4/6-x_5/6, p_18-x_4/6-(1-x_1-x_2-x_3-x_4-x_5)/6, p_19-x_5/6, p_20-x_5/6-(1-x_1-x_2-x_3-x_4-x_5)/6, p_21-(1-x_1-x_2-x_3-x_4-x_5)/6) time I2 = eliminate(J2,{x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5}) time I2a = eliminate({x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5},J2) I2==I2a ****************************************************************** case loaded-loaded J3 = ideal(p_1-x_1*y_1, p_2-x_1*y_2-x_2*y_1, p_3-x_1*y_3-x_3*y_1, p_4-x_1*y_4-x_4*y_1, p_5-x_1*y_5-x_5*y_1, p_6-x_1*(1-y_1-y_2-y_3-y_4-y_5)-(1-x_1-x_2-x_3-x_4-x_5)*y_1, p_7-x_2*y_2, p_8-x_2*y_3-x_3*y_2, p_9-x_2*y_4-x_4*y_2, p_10-x_2*y_5-x_5*y_2, p_11-x_2*(1-y_1-y_2-y_3-y_4-y_5)-(1-x_1-x_2-x_3-x_4-x_5)*y_2, p_12-x_3*y_3, p_13-x_3*y_4-x_4*y_3, p_14-x_3*y_5-x_5*y_3, p_15-x_3*(1-y_1-y_2-y_3-y_4-y_5)-(1-x_1-x_2-x_3-x_4-x_5)*y_3, p_16-x_4*y_4, p_17-x_4*y_5-x_5*y_4, p_18-x_4*(1-y_1-y_2-y_3-y_4-y_5)-(1-x_1-x_2-x_3-x_4-x_5)*y_4, p_19-x_5*y_5, p_20-x_5*(1-y_1-y_2-y_3-y_4-y_5)-(1-x_1-x_2-x_3-x_4-x_5)*y_5, p_21-(1-y_1-y_2-y_3-y_4-y_5)*(1-x_1-x_2-x_3-x_4-x_5)) time I3 = eliminate(J3,{x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5}) ****************************************************************** case fair pair loaded pair J4 = ideal(p_1-1/48-1/4*x_1*y_1, p_2-1/24-(x_1*y_2+x_2*y_1)/4, p_3-1/24-(x_1*y_3+x_3*y_1)/4, p_4-1/24-(x_1*y_4+x_4*y_1)/4, p_5-1/24-(x_1*y_5+x_5*y_1)/4, p_6-1/24-(x_1*(1-y_1-y_2-y_3-y_4-y_5)+(1-x_1-x_2-x_3-x_4-x_5)*y_1)/4, p_7-1/48-1/4*x_2*y_2, p_8-1/24-(x_2*y_3+x_3*y_2)/4, p_9-1/24-(x_2*y_4+x_4*y_2)/4, p_10-1/24-(x_2*y_5+x_5*y_2)/4, p_11-1/24-(x_2*(1-y_1-y_2-y_3-y_4-y_5)+(1-x_1-x_2-x_3-x_4-x_5)*y_2)/4, p_12-1/48-1/4*x_3*y_3, p_13-1/24-(x_3*y_4+x_4*y_3)/4, p_14-1/24-(x_3*y_5+x_5*y_3)/4, p_15-1/24-(x_3*(1-y_1-y_2-y_3-y_4-y_5)+(1-x_1-x_2-x_3-x_4-x_5)*y_3)/4, p_16-1/48-1/4*x_4*y_4, p_17-1/24-(x_4*y_5+x_5*y_4)/4, p_18-1/24-(x_4*(1-y_1-y_2-y_3-y_4-y_5)+(1-x_1-x_2-x_3-x_4-x_5)*y_4)/4, p_19-1/48-1/4*x_5*y_5, p_20-1/24-(x_5*(1-y_1-y_2-y_3-y_4-y_5)+(1-x_1-x_2-x_3-x_4-x_5)*y_5)/4, p_21-1/48-1/4*(1-x_1-x_2-x_3-x_4-x_5)*(1-y_1-y_2-y_3-y_4-y_5)) time I4 = eliminate(J4,{x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5}) ****************************************************************** evaluations m=matrix{{0,0,1/48,1/8,1/16,3/32,0,1/48,1/12,1/24,1/16,1/96,1/24,1/48,5/96,1/48,1/32,13/96,1/96,7/96,3/32,0,0,0,0,0,0,0,0,0,0}} for i from 0 to numgens(I2)-1 list substitute(I2_i,m) for i from 0 to numgens(I3)-1 list substitute(I3_i,m) for i from 0 to numgens(I4)-1 list substitute(I4_i,m) ********************************************************************* Exercise 2 Day 1 restart R=QQ[x_1..x_2,MonomialOrder => Lex] pA=-x_1/10+x_2/50+1/4 pC=2*x_1/25-x_2/100+1/4 pG=11*x_1/100-x_2/50+1/4 pT=-9*x_1/100+x_2/100+1/4 m=matrix{{pA,pC,pG,pT}} L1=diff(x_1,m) L2=diff(x_2,m) m2=matrix{{10*pC*pG*pT},{14*pA*pG*pT},{15*pA*pC*pT},{10*pA*pC*pG}} I=ideal(L1*m2,L2*m2) K= gb I L= gens K L_(0,0) factor (L_(0,0)) f=134456*x_2^3-10852275*x_2^2-4304728125*x_2+935718750 ********************************************************************* Exercise 1 Day 2 restart R=QQ[p_0..p_15,a,b,c,d,e] f1=(p_4+p_11-p_7-p_8)*(p_2+p_13-p_1-p_14)-(p_6+p_9-p_5-p_10)*(p_0+p_15-p_3-p_12) f2=(p_6+p_9-p_3-p_12)*(p_0+p_15-p_5-p_10)-(p_2+p_13-p_7-p_8)*(p_4+p_11-p_1-p_14) f3=(p_3+p_12-p_5-p_10)*(p_0+p_15-p_6-p_9)-(p_1+p_14-p_7-p_8)*(p_2+p_13-p_4-p_11) m=matrix{{f1,f2,f3}} sample=matrix{{1/44,1/44,3/44,7/44,3/44,5/44,1/44,1/44,1/44,1/44,5/44,3/44,7/44,3/44,1/44,1/44,0,0,0,0,0}} substitute(m,sample) J1=ideal(p_0-((1-a)*(1-b)*(1-e)*(1-c)*(1-d)+(1-a)*(1-b)*e*c*d+a*b*e*(1-c)*(1-d)+a*b*(1-e)*c*d),p_1-((1-a)*(1-b)*(1-e)*(1-c)*d+(1-a)*(1-b)*e*c*(1-d)+a*b*e*(1-c)*d+a*b*(1-e)*c*(1-d)),p_2-((1-a)*(1-b)*(1-e)*c*(1-d)+(1-a)*(1-b)*e*(1-c)*d+a*b*e*c*(1-d)+a*b*(1-e)*(1-c)*d),p_3-((1-a)*(1-b)*(1-e)*c*d+(1-a)*(1-b)*e*(1-c)*(1-d)+a*b*e*c*d+a*b*(1-e)*(1-c)*(1-d)),p_4-((1-a)*b*(1-e)*(1-c)*(1-d)+(1-a)*b*e*c*d+a*(1-b)*e*(1-c)*(1-d)+a*(1-b)*(1-e)*c*d),p_5-((1-a)*b*(1-e)*(1-c)*d+(1-a)*b*e*c*(1-d)+a*(1-b)*e*(1-c)*d+a*(1-b)*(1-e)*c*(1-d)),p_6-((1-a)*b*(1-e)*c*(1-d)+(1-a)*b*e*(1-c)*d+a*(1-b)*e*c*(1-d)+a*(1-b)*(1-e)*(1-c)*d),p_7-((1-a)*b*(1-e)*c*d+(1-a)*b*e*(1-c)*(1-d)+a*(1-b)*e*c*d+a*(1-b)*(1-e)*(1-c)*(1-d)),p_8-((1-a)*b*(1-e)*c*d+(1-a)*b*e*(1-c)*(1-d)+a*(1-b)*e*c*d+a*(1-b)*(1-e)*(1-c)*(1-d)),p_9-((1-a)*b*(1-e)*c*(1-d)+(1-a)*b*e*(1-c)*d+a*(1-b)*e*c*(1-d)+a*(1-b)*(1-e)*(1-c)*d),p_10-((1-a)*b*(1-e)*(1-c)*d+(1-a)*b*e*c*(1-d)+a*(1-b)*e*(1-c)*d+a*(1-b)*(1-e)*c*(1-d)),p_11-((1-a)*b*(1-e)*(1-c)*(1-d)+(1-a)*b*e*c*d+a*(1-b)*e*(1-c)*(1-d)+a*(1-b)*(1-e)*c*d),p_12-((1-a)*(1-b)*(1-e)*c*d+(1-a)*(1-b)*e*(1-c)*(1-d)+a*b*e*c*d+a*b*(1-e)*(1-c)*(1-d)),p_13-((1-a)*(1-b)*(1-e)*c*(1-d)+(1-a)*(1-b)*e*(1-c)*d+a*b*e*c*(1-d)+a*b*(1-e)*(1-c)*d),p_14-((1-a)*(1-b)*(1-e)*(1-c)*d+(1-a)*(1-b)*e*c*(1-d)+a*b*e*(1-c)*d+a*b*(1-e)*c*(1-d)),p_15-((1-a)*(1-b)*(1-e)*(1-c)*(1-d)+(1-a)*(1-b)*e*c*d+a*b*e*(1-c)*(1-d)+a*b*(1-e)*c*d)) J2=ideal(p_0-((1-a)*(1-b)*(1-e)*(1-c)*(1-d)+(1-a)*b*e*(1-c)*d+a*(1-b)*e*c*(1-d)+a*b*(1-e)*c*d),p_1-((1-a)*(1-b)*(1-e)*(1-c)*d+(1-a)*b*e*(1-c)*(1-d)+a*(1-b)*e*c*d+a*b*(1-e)*c*(1-d)),p_2-((1-a)*(1-b)*(1-e)*c*(1-d)+(1-a)*b*e*c*d+a*(1-b)*e*(1-c)*(1-d)+a*b*(1-e)*(1-c)*d),p_3-((1-a)*(1-b)*(1-e)*c*d+(1-a)*b*e*c*(1-d)+a*(1-b)*e*(1-c)*d+a*b*(1-e)*(1-c)*(1-d)),p_4-((1-a)*b*(1-e)*(1-c)*(1-d)+(1-a)*(1-b)*e*(1-c)*d+a*b*e*c*(1-d)+a*(1-b)*(1-e)*c*d),p_5-((1-a)*b*(1-e)*(1-c)*d+(1-a)*(1-b)*e*(1-c)*(1-d)+a*b*e*c*d+a*(1-b)*(1-e)*c*(1-d)),p_6-((1-a)*b*(1-e)*c*(1-d)+(1-a)*(1-b)*e*c*d+a*b*e*(1-c)*(1-d)+a*(1-b)*(1-e)*(1-c)*d),p_7-((1-a)*b*(1-e)*c*d+(1-a)*(1-b)*e*c*(1-d)+a*b*e*(1-c)*d+a*(1-b)*(1-e)*(1-c)*(1-d)),p_8-((1-a)*b*(1-e)*c*d+(1-a)*(1-b)*e*c*(1-d)+a*b*e*(1-c)*d+a*(1-b)*(1-e)*(1-c)*(1-d)),p_9-((1-a)*b*(1-e)*c*(1-d)+(1-a)*(1-b)*e*c*d+a*b*e*(1-c)*(1-d)+a*(1-b)*(1-e)*(1-c)*d),p_10-((1-a)*b*(1-e)*(1-c)*d+(1-a)*(1-b)*e*(1-c)*(1-d)+a*b*e*c*d+a*(1-b)*(1-e)*c*(1-d)),p_11-((1-a)*b*(1-e)*(1-c)*(1-d)+(1-a)*(1-b)*e*(1-c)*d+a*b*e*c*(1-d)+a*(1-b)*(1-e)*c*d),p_12-((1-a)*(1-b)*(1-e)*c*d+(1-a)*b*e*c*(1-d)+a*(1-b)*e*(1-c)*d+a*b*(1-e)*(1-c)*(1-d)),p_13-((1-a)*(1-b)*(1-e)*c*(1-d)+(1-a)*b*e*c*d+a*(1-b)*e*(1-c)*(1-d)+a*b*(1-e)*(1-c)*d),p_14-((1-a)*(1-b)*(1-e)*(1-c)*d+(1-a)*b*e*(1-c)*(1-d)+a*(1-b)*e*c*d+a*b*(1-e)*c*(1-d)),p_15-((1-a)*(1-b)*(1-e)*(1-c)*(1-d)+(1-a)*b*e*(1-c)*d+a*(1-b)*e*c*(1-d)+a*b*(1-e)*c*d)) J3=ideal(p_0-((1-a)*(1-b)*(1-e)*(1-c)*(1-d)+(1-a)*b*e*c*(1-d)+a*(1-b)*e*(1-c)*d+a*b*(1-e)*c*d),p_1-((1-a)*(1-b)*(1-e)*(1-c)*d+(1-a)*b*e*c*d+a*(1-b)*e*(1-c)*(1-d)+a*b*(1-e)*c*(1-d)),p_2-((1-a)*(1-b)*(1-e)*c*(1-d)+(1-a)*b*e*(1-c)*(1-d)+a*(1-b)*e*c*d+a*b*(1-e)*(1-c)*d),p_3-((1-a)*(1-b)*(1-e)*c*d+(1-a)*b*e*(1-c)*d+a*(1-b)*e*c*(1-d)+a*b*(1-e)*(1-c)*(1-d)),p_4-((1-a)*b*(1-e)*(1-c)*(1-d)+(1-a)*(1-b)*e*c*(1-d)+a*b*e*(1-c)*d+a*(1-b)*(1-e)*c*d),p_5-((1-a)*b*(1-e)*(1-c)*d+(1-a)*(1-b)*e*c*d+a*b*e*(1-c)*(1-d)+a*(1-b)*(1-e)*c*(1-d)),p_6-((1-a)*b*(1-e)*c*(1-d)+(1-a)*(1-b)*e*(1-c)*(1-d)+a*b*e*c*d+a*(1-b)*(1-e)*(1-c)*d),p_7-((1-a)*b*(1-e)*c*d+(1-a)*(1-b)*e*(1-c)*d+a*b*e*c*(1-d)+a*(1-b)*(1-e)*(1-c)*(1-d)),p_8-((1-a)*b*(1-e)*c*d+(1-a)*(1-b)*e*(1-c)*d+a*b*e*c*(1-d)+a*(1-b)*(1-e)*(1-c)*(1-d)),p_9-((1-a)*b*(1-e)*c*(1-d)+(1-a)*(1-b)*e*(1-c)*(1-d)+a*b*e*c*d+a*(1-b)*(1-e)*(1-c)*d),p_10-((1-a)*b*(1-e)*(1-c)*d+(1-a)*(1-b)*e*c*d+a*b*e*(1-c)*(1-d)+a*(1-b)*(1-e)*c*(1-d)),p_11-((1-a)*b*(1-e)*(1-c)*(1-d)+(1-a)*(1-b)*e*c*(1-d)+a*b*e*(1-c)*d+a*(1-b)*(1-e)*c*d),p_12-((1-a)*(1-b)*(1-e)*c*d+(1-a)*b*e*(1-c)*d+a*(1-b)*e*c*(1-d)+a*b*(1-e)*(1-c)*(1-d)),p_13-((1-a)*(1-b)*(1-e)*c*(1-d)+(1-a)*b*e*(1-c)*(1-d)+a*(1-b)*e*c*d+a*b*(1-e)*(1-c)*d),p_14-((1-a)*(1-b)*(1-e)*(1-c)*d+(1-a)*b*e*c*d+a*(1-b)*e*(1-c)*(1-d)+a*b*(1-e)*c*(1-d)),p_15-((1-a)*(1-b)*(1-e)*(1-c)*(1-d)+(1-a)*b*e*c*(1-d)+a*(1-b)*e*(1-c)*d+a*b*(1-e)*c*d)) time I1 = eliminate(J1,{a,b,c,d,e}) time I2 = eliminate(J2,{a,b,c,d,e}) time I3 = eliminate(J3,{a,b,c,d,e}) for i from 0 to numgens(I1)-1 list substitute(I2_i,sample) for i from 0 to numgens(I2)-1 list substitute(I3_i,sample) for i from 0 to numgens(I3)-1 list substitute(I3_i,sample) I1 == I2 I1 == I3 I2 == I3 f1 % I1 == 0 f2 % I1 == 0 f3 % I1 == 0 f1 % I2 == 0 f2 % I2 == 0 f3 % I2 == 0 f1 % I3 == 0 f2 % I3 == 0 f3 % I3 == 0