Kahan’s ellipse problem (SIGSAM Bulletin, 9, 11, 1975) 楕円周が単位開円板に含まれるための条件a^2＞0 ∧ b^2＞0 ∧ ∀x∀y({(x-c)^2}/{a^2}+{(y-d)^2}/{b^2}＝1 → x^2+y^2＜1) を a^2，b^2，c^2，d^2 についての多項式による不等式の連言で表すには少なくと…

julia> using GroebnerBasis, Singular julia> R, (a, b, c) = PolynomialRing(QQ, ["a", "b", "c"]); julia> I(n) = Ideal(R, [(a+b)^n+c-3, (b+c)^n+a-2, (2*c+a)^n+b-1]); julia> for n = 5:15 print(n,":"); @time msolve(I(n), get_param = true) end; …

_ _ _(_)_ | Documentation: https://docs.julialang.org (_) | (_) (_) | _ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help. | | | | | | |/ _` | | | | |_| | | | (_| | | Version 1.6.0 (2021-03-24) _/ |\__'_|_|_|\__'_| | Official https://jul…

? tst12([all,all],[b,a,c,x,y],(f1,f2,f3,f4)->f1*f2*imp(f3,f4),"0

2017年に続いて，2020年に発表された Brown と McCallum の論文 "Enhancements to Lazard’s Method for Cylindrical Algebraic Decomposition" https://rdcu.be/ciEY8 の主結果は次の通りです． Lazard method の projection set に trailing coefficient を…

Scott McCallum http://www.casc-conference.org/2020/videos/1/01-BrownMcCallum.mp4Chris Brown, Hoon Hong, James Davenport https://vimeo.com/showcase/5271198 http://www.sc-square.org/CSA/school/lectures.htmlAdam Strzebonski https://youtu.be/z…

? tst12([all,all,all],[a,b,c,d,e,f,x,y,z],id,"((a*x+b)*y+c)*z+((d*x+e)*y+f)<0");Ans() *** using Lazard's method (MPP17). [z,1] [y,2] [x,3] [f,3] [e,2] [d,1] [c,1] [b,1] [a,1] time = 74 ms. 3 9(0,0) 27(0,0) 81(0,0) 291(0,60) 1257(0,396) 580…

? tst12([],[c,a,d,b],(f1,f2,f3,f4,f5)->imp(f1*f2*f3*f4,f5),"a^2<=a,b^2<=b,c^2<=c,d^2<=d,(1-a^2*b^2)*(1-c*d)*(a*d-b*c)^2+2*a*b*(c*d-a*b)*(1-a*b)*(c-d)^2+(a^2*b^2-c^2*d^2)*(1-c*d)*(a-b)^2>=0",17);Ans(); *** using Lazard's method (BM20). 1 [b…

? tst12([all,ex],[a,b,x,y],(f1,f2)->f1*f2,"x^6+x*y+a>0,x+b*y^6+a<0",17);Ans() *** using Lazard's method (BM20). 1,0 [y,2] -5 [x,3] -5 [b,2] [a,2] [[[a],[],[x],[]]] time = 1718 ms. 5 25(0,2) 180(55,29) 113(108,74) *** combined adjacent 88 c…

? tst12([all,all],[a,b,c,x,y],(f1,f2)->imp(f1,f2),"x^8+y^8<1,a*x+b*y+c<0"); *** using Lazard's method (MPP17). [y,2] [x,4] [c,10] [b,12] [a,1] time = 14144 ms. 3 65(206,12) 963(196,62) 9219(384,432) 4385(8672,876) *** combined adjacent 437…

? tst12([all,all,all],[a,b,c,d,x,y,z],(f1,f2)->imp(f1,f2),"x^2+y^2+z^2<1,a*x+b*y+c*z+d<0"); *** using Lazard's method (MPP17). [z,2] [y,3] [x,6] [d,11] [c,11] [b,8] [a,1] time = 284 ms. 3 33(10,8) 499(36,111) 12849(20,2411) 183385(6996,102…

? tst12([],[a,b,c,x,y],f->f,"(a*x^2+b*x-c)*y^2+(b*x^2+c*x-a)*y+c*x^2+a*x-b==0"); *** using Lazard's method (MPP17). [y,1] [x,3] [c,11] [b,41] [a,1] time = 414 ms. 3 217(188,41) 5631(66,259) 64685(1784,2346) 89867(26164,0) *** combined adja…

Mathematica による結果の検証． ? tst12([],[a,b,c,d,e,f,x],(f1,f2,f3,f4,f5,f6,f7)->f1*f2*f3*f4*f5*f6*f7,"a==0,b<0,c<0,d==0,e<0,a*x^2+b*x+c>0,d*x^2+e*x+f<0",11);Ans(1) *** roots will be displayed in [pol,[zero,root,multi,deg]] for mma. *** u…

? tst12([],[a,b,c,x],f1->f1,"a*x^2+(a+b)*x+b*c<1",17); *** using Lazard's method (BM20). -5 [x,1] 0 [c,1] [b,1] [a,1] [[[a],[b],[],[]],[[a+4],[b],[],[]]] time = 264 ms. 5 15(0,0) [[a,b],[-4,0]]:-a^2+((4*c-2)*b-4)*a-b^2>>(4*c-2)*a-2*b 33(0,…

? tst12([],[a,b,c,x,y],(f1,f2)->f1*f2,"y>=0,y+(a-b)*x^2+(2*a-3*b+c)*a*x+c-b<=0");pp *** using Lazard's method (MPP17). [y,2] [x,1] [c,2] [b,2] [a,4] time = 134 ms. 11 57(6,2) 315(36,26) 1219(148,0) 1592(0,946) *** combined adjacent 786 cel…