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Question: How can I understand and work the Output of RegularChains?

Question:How can I understand and work the Output of RegularChains?

Saalehorizontale 35 Maple 2024
solve regularchains
January 08 2025
3

Hey guys,

I have to solve a bunch of systems of polynomial equations und dome restrictions given by inequalitites. I have 8 variables, 8 equations and and 13 inequalitites. Since the simple solve or SemiAlgebraic command are not able to solve every system I tryd some other ways. Right now I try to bring the set of equations and ineqaulities in a better from or structure using RealTriangulize from the RegularChains library. Later on I want to take those results and use solve or SemiAlgebraic again, hoping, that Maple than finds the solutions and is not calculating for houres without a result. I already know, that you can have diffrent outputs for RealTriangularize (I know list, record, piecewise and zerodimensional, althought the last one is not really helpful). Since I want to go on wirking with the results I need to have them in a form, that I can read of the new equations and inequalities to put them into solve. Often that works totaly fine, but sometimes I get an output I dont understand. I understand what It means but I dont understand why Maple uses that type of output. If you have a look in the attached file you can see what I mean:

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restart; with(RegularChains); eq_5334 := {y*(m*x-m-n+1)+(-x+1)*n-x = 0, (-p+t)*k+p*y-t = 0, (k-x-y)*t-k*p+y = 0, (-x-y+1)*t+(-k+y)*n+x*s = 0, (-x-y+1)*p+m*y^2+x-y = 0, (x^2-x)*m+y*(t-1)-n+1 = 0, -k*n+s*x = 0, m*x*y-p = 0, 0 < k, 0 < m, 0 < s, 0 < x, 0 < y, 0 < n+(t-1)*p, 0 < (m*y-1)*n+(1-p)*(m*x-m+1), 0 < (m*x-m-t+1)*p+m*y*(t-n), 1 < x+y, k < 1, m < 1, s < t, t < 1}; eq_5380 := {(-x-y+1)*p+m*x*y = 0, (-p+t)*k+p*y-t = 0, (k-x-y)*t-k*p+y = 0, (-x-y+1)*t+(-k+y)*n+x*s = 0, (m-1)*y^2+(-x+1)*y-p+x = 0, (x-1)*(m-1)*y-x^2-n+x = 0, m*x^2+(-m-n+1)*x+(-y+1)*n+t*y-1 = 0, -k*n+s*x = 0, 0 < k, 0 < m, 0 < s, 0 < x, 0 < y, 0 < n+(t-1)*p, 0 < (m*y-1)*n+(1-p)*(m*x-m+1), 0 < (m*x-m-t+1)*p+m*y*(t-n), 1 < x+y, k < 1, m < 1, s < t, t < 1}; eq_5382 := {(-x-y+1)*p+m*x*y = 0, y*(m*x-m-n+1)+(-x+1)*n-x = 0, (-p+t)*k+p*y-t = 0, (k-x-y)*t-k*p+y = 0, (-x-y+1)*t+(-k+y)*n+x*s = 0, (-x-y+1)*p+m*y^2+x-y = 0, m*x^2+(-m-n+1)*x+(-y+1)*n+t*y-1 = 0, -k*n+s*x = 0, 0 < k, 0 < m, 0 < s, 0 < x, 0 < y, 0 < n+(t-1)*p, 0 < (m*y-1)*n+(1-p)*(m*x-m+1), 0 < (m*x-m-t+1)*p+m*y*(t-n), 1 < x+y, k < 1, m < 1, s < t, t < 1}; sys := eq_5334; SuggestVariableOrder(sys); R := PolynomialRing(%); dec_5334 := RealTriangularize(sys, R, output = piecewise); sys := eq_5380; SuggestVariableOrder(sys); R := PolynomialRing(%); dec_5380 := RealTriangularize(sys, R, output = piecewise); sys := eq_5382; SuggestVariableOrder(sys); R := PolynomialRing(%); dec_5382 := RealTriangularize(sys, R, output = piecewise); sys := eq_5382; SuggestVariableOrder(sys); R := PolynomialRing(%); dec_5382_record := RealTriangularize(sys, R, output = record)

[AlgebraicGeometryTools, ChainTools, ConstructibleSetTools, Display, DisplayPolynomialRing, Equations, ExtendedRegularGcd, FastArithmeticTools, Inequations, Info, Initial, Intersect, Inverse, IsRegular, LazyRealTriangularize, MainDegree, MainVariable, MatrixCombine, MatrixTools, NormalForm, ParametricSystemTools, PolynomialRing, Rank, RealTriangularize, RegularGcd, RegularizeInitial, SamplePoints, SemiAlgebraicSetTools, Separant, SparsePseudoRemainder, SuggestVariableOrder, Tail, Triangularize]

  

[s, k, n, p, m, t, x, y]

  

R := polynomial_ring

  

dec_5334 := [[x*s+((-x^2+x)*m-t*y+y-1)*k = 0, (m*x*y-t)*k+(x+y)*t-y = 0, n+(-x^2+x)*m-t*y+y-1 = 0, -m*x*y+p = 0, (x^2*y+(y^2-y)*x-y^2)*m-x+y = 0, t*y^2-y^2+x = 0, (15*y^2+24*y+20)*x-6*y^2-13*y-10 = 0, y^3-y-2 = 0, 0 < k, 0 < m, 0 < s, 0 < x, 0 < y, 0 < 12891634966*y^2+19613071879*y+16947294542, 0 < 1256597*y^2+1911761*y+1651926, 0 < 6310892468*y^2+9601263717*y+8296275330, 0 < 1401*y^2+2130*y+1840, 0 < 1-k, 0 < 1-m, 0 < 72927541996846438*y^2+110950482461140595*y+95870270479707846, 0 < 1-t]]

  

[s, k, n, p, m, t, y, x]

  

R := polynomial_ring

  

dec_5380 := piecewise(`and`(`and`(`and`(0 < x^3-2*x^2+3*x-1, 0 < x^3+2*x^2+x-1), x^3+x^2+x < 1), 0 < 3*x-1), [[s*x+((1-x)*y*m+(x-1)*y+x^2-x)*k = 0, (m*y^2-y^2-t+(1-x)*y+x)*k+(y+x)*t-y = 0, n+(1-x)*y*m+(x-1)*y+x^2-x = 0, p-m*y^2+y^2+(x-1)*y-x = 0, m*y-x-y+1 = 0, t*y^2+(x-1)*y^2+(2*x^2-2*x)*y+x^3-2*x^2+x = 0, (3*x-1)*y^2+(3*x^2-3*x)*y+x^3-2*x^2+x = 0, 0 < k, 0 < m, 0 < s, 0 < y, 0 < -6*x^6-9*x^5*y+20*x^5+27*x^4*y-27*x^4-32*x^3*y+19*x^3+17*x^2*y-7*x^2-3*x*y+x, 0 < 3*x^6+3*x^5*y-14*x^5-10*x^4*y+26*x^4+11*x^3*y-24*x^3-3*x^2*y+11*x^2-2*x*y-2*x+y, 0 < 6*x^5+9*x^4*y-17*x^4-18*x^3*y+17*x^3+11*x^2*y-7*x^2-2*x*y+x, 0 < y+x-1, 0 < 1-k, 0 < -m+1, 0 < t-s, 0 < 1-t]], [])

  

[s, k, n, p, m, t, x, y]

  

R := polynomial_ring

  

dec_5382 := piecewise(`and`(`and`(y^3-2*y^2+y < 1, 0 < y-1), 23*y^3-37*y^2+13*y-3 <> 0), [[-k*n+s*x = 0, (p-t)*k+(y+x)*t-y = 0, (y+x-1)*n+(-x*y+y)*m+x-y = 0, (y+x-1)*p-m*y^2-x+y = 0, m*y-1 = 0, t*y^2+x^2+(y-1)*x-y^2 = 0, x^3+(3*y-2)*x^2+(2*y^2-3*y+1)*x-y^3+y^2 = 0, 0 < k, 0 < s, 0 < x, 0 < -2*x^2*y^2-2*x*y^3+2*y^4+x^2*y+3*x*y^2-3*y^3-x*y+y^2, 0 < x^2*y^2+2*x*y^3+y^4-x^2*y-4*x*y^2-3*y^3+2*x*y+3*y^2-y, 0 < -x^2*y-x*y^2+y^3+x*y-y^2, 0 < y+x-1, 0 < 1-k, 0 < t-s, 0 < 1-t]], 23*y^3-37*y^2+13*y-3 = 0, [[-k*n+s*x = 0, (p-t)*k+(y+x)*t-y = 0, (y+x-1)*n+(-x*y+y)*m+x-y = 0, (y+x-1)*p-m*y^2-x+y = 0, m*y-1 = 0, t*y^2+x^2+(y-1)*x-y^2 = 0, (2377326*y^2-1587000*y+302588)*x^2+(390793*y^2+497766*y+138115)*x-507805*y^2+152032*y-109047 = 0, 23*y^3-37*y^2+13*y-3 = 0, 0 < k, 0 < m, 0 < s, 0 < x, 0 < y, 0 < 700112222844255556263586865*x*y^2-260269572171898884295316974*x*y-93795749047261033657544191*y^2+73822886321394794237709987*x+34866975665513154551125606*y-9877974587657378842117575, 0 < -26166721441919*x*y^2+9412709182291*x*y+53422638514257*y^2-3387596446782*x-21180373503698*y+6484087812711, 0 < 21236600258115*x*y^2-8079468597142*x*y-3053799376681*y^2+2340822678357*x+1387037467490*y-370794765921, 0 < y+x-1, 0 < 1-k, 0 < -m+1, 0 < t-s, 0 < 1-t]], [])

  

[s, k, n, p, m, t, x, y]

  

R := polynomial_ring

  

`Non-fatal error while reading data from kernel.`

 (1)
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NULL

I would like to get results like in dec_5334. I can easily go on working with this kind of form. In dec_5380 you can see a diffrent output. I dont see the point of giving me this output. the second line i basically epmty. and in the first line the solution is broken into peaces. when a certain solution just works under some inequalitites, why dont they put those four inequalities inside of the list in front of it? Is there a workaround for the "normal" output? Or is there a way to read off the lines from this kind of structure, with the open { in front ?

The same problem appears in dec_5382. WHy dont give me a list with to lists of equations and inequalities to show me both solutions?
In the last example dec_5382_record you can see the output when you change the corresponding option in RealTrinagularize. But here I again have the problem that I dont know how to read of the equations and inequalities from the open curly bracket.

If anyone could help me, I would be very glad. Thank yu in advance.

Regards

Felix

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