Computing the real solutions of polynomial systems with the RegularChains library in Maple Changbo Chen1 , James H. Davenport2, Francois Lemaire3, Marc Moreno Maza1, Bican Xia4, Rong Xiao1, Yuzhen Xie1 1University of Western Ontario, Canada; 2University of Bath, UK; 3Universit\303\251 de Lille, France; 4Peking University, China

The RegularChains library offers a variety of tools to compute the real solutions of polynomial systems. These tools include isolating/counting the real solutions of zero-dimensional systems, describing real solutions of positive dimensional systems, classifing the number of real roots of parametric systems, finding sample points (thus determining emptiness) of semi-algebraic sets, performing set theoretical operations on semi-algebraic sets as well as computing cylindrical algebraic decompositions. The theory and algorithms underlying these tools are described in [10, 11, 5, 1, 3, 4]. Most of the commands implementing these tools are part of the SemiAlgebraicSetTools module while the others can be found in the ParametricSystemTools module or at the top level of the RegularChains library.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEkTGV0RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSJ+RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GLDYmUSJTRidGL0YyRjVGL0Y8 be the semi-algebraic system that the real solving commands operating on. In most situations, it is encoded by 4 lists of polynomials, respectively for equations, non-negative inequalities, positive inequalities, and inequations.

Isolate the real solutions of a simple system { x^2+y^2-1=0, x*y-1/3=0, x-y>0 }. R := PolynomialRing([y,x]): F := [x^2+y^2-1, x*y-1/3]: N := []: P := [x-y]: H := []: sols := RealRootIsolate(F, N, P, H, R); Show the isolating intervals. map(BoxValues, sols, R); Refine boxes to a given absolute precision. sols_abserr := map(RefineBox, sols, 1/2^10, R): map(BoxValues, sols_abserr, R); Refine boxes to a given relative precision. sols_rerr := map(RefineBox, sols, errtype='relative', 1/2^10, R): map(BoxValues, sols_rerr, R);

Consider the generic equation of degree two. R := PolynomialRing([x, c, b, a]); sys := [a*x^2+b*x+c=0]; Compute a triangular decomposition of the 4-variable hypersurface it defines. dec := RealTriangularize(sys, R); Display(dec, R); Consider the record output format. RealTriangularize(sys, R, output=record); Next, we consider an example from Brown and McCallum's ISSAC paper. Consider the following question: when does 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 have a non-real root LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiK0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk8tRiw2JlEjaXlGJ0YvRjJGNS1GOTYuUSJ+RidGL0Y8Rj5GQUZDRkVGR0ZJRksvRk5RJjAuMGVtRicvRlFGWUY8satisfying LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEjeHlGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIjxGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2JVEiMUYnRi9GPC1JJm10ZXh0R0YkNiVRIz9+RidGL0Y8LUYsNiNRIUYnRjw=This problem can be expressed as a quantifier elimination problem: 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 Next we call RealTriangularize to compute the semi-algebraic system 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 JSFH R := PolynomialRing([y, x, b, a]); JSFH JSFH JSFH sys := [x^3-3*y^2*x+a*x+b=0, 3*x^2-y^2+a=0, 1-y*x>0, y<>0]; JSFH JSFH out := RealTriangularize(sys, R, output=record); JSFH From the output, one could easily see that the answer of the QE problem is: 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 Consider now an example which has finitely many complex solutions. R := PolynomialRing([x, y, z]): Define a set of equations. sys := [x^3 + y + z -1=0, x + y^3 + z -1=0, x + y + z^3 -1=0]; Compute the real solutions of sys. dec := RealTriangularize(sys, R, output=record); Returning now to systems which have infinitely many complex solutions, consider the intersection of the algebraic surfaces Sofa and Cylinder from the Algebraic Surface Gallery. As their names suggest, they are respectively the equations of a sofa and a (sort of) cylinder. One expects to find a real curve in their intersection, as we shall verify. R := PolynomialRing([z, y, x]); Sofa := x^2 + y^3 + z^5; Cyl := x^4 + z^2 - 1; The plot of the two surfaces. plots:-implicitplot3d([Sofa=0, Cyl=0], x=-5..5, y=-5..5, z=-5..5, color=[red, blue], numpoints=400000); Compute the triangualr decomposition of intersection of the two surfaces. RealTriangularize([Sofa, Cyl], R, output=record);

Consider again the generic equation of degree two. Now we solve it interactively. R := PolynomialRing([x, c, b, a]); sys := [a*x^2+b*x+c=0]; Use LazyRealTriangularize to start the decomposition. dec := LazyRealTriangularize(sys, R); Go one step further, computing components in lower dimension. dec2 := value(dec); Go the last step, computing components in dimension zero. value(dec2); If one is only interested in computing the main components, the list output format does the job. dec := LazyRealTriangularize(sys, R, output=list); Display(dec, R); Another way to conduct the computation interactively is to use the record output format. dec := [LazyRealTriangularize(sys, R, output=record)]; Go one step further. dec2 := value(dec); Go the last step. value(dec2); Computing a lazy triangular decomposition is usually much less expensive than computing a full one. The following is an example. variables := [x, u, v, w]; R := PolynomialRing(variables); sys := [u*x^2+v*x+1=0, v*x^3+w*x+u=0, w*x^2+v*x+u<=0]; Computing a lazy decomposition takes than a second. dec := LazyRealTriangularize(sys,R,output=list); Computing a full one does not terminate within an hour.

Consider again the generic equation of degree two. R := PolynomialRing([x, c, b, a]); F := [a*x^2+b*x+c=0]; Compute sample points of the 4-variable hypersurface it defines. S := SamplePoints(F, R, output=record); Consider the Tacnode curve. We look for sample points in the middle of the right branch. R := PolynomialRing([y, x]); F := [y^4-2*y^3+y^2-3*x^2*y+2*x^4]; with(plots): implicitplot(F, x=-2..2, y=-1..3, numpoints=1000000); SamplePoints([op(F), 2*x > 1, x < 1], R, output=record);

Consider a simple system {x^3-a*x+b=0, x>0}. R := PolynomialRing([x,a,b]); rrc := RealRootClassification([x^3-a*x+b], [], [x], [], 2, 1, R); We can use Display command to get the user-friend information for the typed objects. Show the quantifier free formula encoded by the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE7cmVndWxhcl9zZW1pX2FsZ2VicmFpY19zZXRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGL0Yy rsas := rrc[1][1]: Display(rsas, R); Show the contend of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEyYm9yZGVyX3BvbHlub21pYWxGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGL0Yy bp := rrc[2]: Display(bp, R); One can continue investigating the parameter values vanishing the border polynomial factors. rrc_b := RealRootClassification([x^3-a*x+b,b], [], [x], [], 2, 1, R): Display(rrc_b[1][1],R); Display(rrc_b[2],R);

S is the semi-algebraic set defined by a*x^2+b*x+c = 0. R := PolynomialRing([x,a,b,c]): S := RealTriangularize([a*x^2+b*x+c = 0], R): map(Display,S,R); A is the semi-algebraic set defined by a=0. A := RealTriangularize([a = 0], R): map(Display,A,R); Compute the difference of S \134 A. S_A := Difference(S,A,R): map(Display,S_A,R);

R := PolynomialRing([y,x]): cad := CylindricalAlgebraicDecompose([x*y-1], R);

The following two systems S1 and S2 are both conditions for a triangle with edge lengths a, b, c (denoting the respective edges a, b, c too) to satisfy that the external bisector of the angle of a, c intersects with b on the other side of the triangle. We set S1 up first. C1:=[a > 0 , b > 0 , c > 0 , a < b + c , b < a + c , c < a + b, b^2 + a^2 - c^2 <= 0 ]: C2:=[a > 0 , b > 0 , c > 0 , a < b + c , b < a + c , c < a + b, c*(b^2 + a^2 - c^2)^2 < a*b^2*(2*a*c - (c^2 + a^2 - b^2))]: S1 is the disjuction of C1 and C2. S1:=[C1, C2]; Then we set S2. S2 := [a-c<0, a > 0 , b > 0 , c > 0 , a < b + c , b < a + c , c < a + b]; Compute regular semi-algebraic system representations dec1 (resp. dec2) for S1 (resp. S2) R := PolynomialRing([a,b,c]): dec1 := map(op, map(RealTriangularize, S1,R)); dec2:=RealTriangularize(S2, R); Compute the differences: S1 \134 S2 and S2 \134 S1 Difference(dec1,dec2,R); Difference(dec2,dec1,R);

In any 2D affine space, "3 points A1, A2, A3 are colinear" can be characterized by the following condition. colinear := (A1,A2,A3) -> (A1[1]-A2[1])*(A2[2]-A3[2])-(A1[2]-A2[2])*(A2[1]-A3[1])=0: "3 points A1, A2, A3 are not colinear" can be characterized by the following condition. non_colinear := (A1,A2,A3) -> (A1[1]-A2[1])*(A2[2]-A3[2])-(A1[2]-A2[2])*(A2[1]-A3[1])<>0: In \342\204\2352 (orientable), suppose we have the following set of points. 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The above picture of 9 points in \342\204\2352 satisfies: a. (alignment) 123, 456, 186, 175, 274, 296, 384, 395 are colinear; b. (orientation) 9 is above 456 but below 384, 7, 1 and 2 are above both 456 and 384. The non-Pappus matroid realization problem is to find a set of 9 points satisfying the above items a. b., but 789 are not colinear. We use the fact that 456, 384 form two intersecting lines to build a coordinate system. The coordinates are set below: A1 := [g,h]; A2 := [k,l]; A3 := [0,1]; A4 := [0,0]; A5 := [a,0]; A6 := [1,0]; A7 := [c,d]; A8 := [0,b]; A9 := [e,f]; Set the constraints of alignments. colinears := [colinear(A1,A2,A3), colinear(A1,A7,A5), colinear(A1,A8,A6),colinear(A2,A7,A4),colinear(A2,A9,A6), colinear(A3,A9,A5)]; Set the constraints of non-alignments. non_colinears := [non_colinear(A7,A8,A9)]; Set the constraints of orientation. orientation := [e>0, f>0, c<0, d>0, g<0, h>0]; Put all the above constraints together. cons:=[op(colinears),op(non_colinears),op(orientation)]; We use SamplePoints to find realization points of the above constrains, empyt implies no possible realization of non-Pappus matroid. vars:=indets(cons): R:=PolynomialRing(convert(vars,list)): ss:=SamplePoints(cons,R);

Consider the following biochemistry reaction network. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= --k1--> E + S C <-km1-- E (Enzyme) and S (Substrate) reacting together producing C (Complex). k1, km1 are reaction rate constants. Question: Is there a unique positive equilibrium? First build the chemical model. 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 JSFH We generate the dynamical system governing the reaction. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2KlEzUmVhY3Rpb25TeXN0ZW0yT0RTRicvJSdmYW1pbHlHUStNb25vc3BhY2VkRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUSpbMjU1LDAsMF1GJy8lMGZvbnRfc3R5bGVfbmFtZUdRLE1hcGxlfklucHV0RicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkAtSShtZmVuY2VkR0YkNiktRiM2Ki1GLDYqUSNSU0YnRi9GMkY1RjhGO0Y+RkEtSSNtb0dGJDYyUSIsRidGL0YyRjhGO0Y+RkEvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYqUSpTdWJzT3JkZXJGJ0YvRjJGNUY4RjtGPkZBRi9GMi8lK2V4ZWN1dGFibGVHRjdGPkZBRi9GMkY4RjtGPkZBLUZMNjJRIjtGJ0YvRjJGOEY7Rj5GQUZPRlFGU0ZVRldGWUZlbkZnbi9GW29RLDAuMjc3Nzc3OGVtRidGL0YyRmBvRj5GQQ== We obtain the equations defining the equilibria and the conversation laws. Together they form a system of non-linear algebraic equations. Here, C + S - C_0 - S_0 means that C(t) + S(t) is constant along the time. Eqs := Equilibria(RS); Claws := ConservationLaws(RS); F := [ op(Claws), op(Eqs) ]; JSFH The current concentrations (given by the substrate names) are the unknowns. The rate constants and initial concentrations are regarded as parameters. JSFH rates := GetAllRateConstants(RS): current := GetAllSubstratesName(RS): init := map(u->cat(u,"_0"), GetAllSubstratesName(RS)): allvars := [op(current), op(init), op(rates)]; R := PolynomialRing(allvars); N := []; P := allvars; H := []; JSFH We compute a triangular decomposition of the semi-algebraic system defined by 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 JSFH dec := RealTriangularize(F, N, P, H, R); JSFH The system has at least one positive equilibrium if and only if the above contraints hold Display(dec[1], R); JSFH Extract the regular chain, quantifier-free forumula and positive inqualities of the computed regular semi-algebraic system. rsas := dec[1]; rc := RepresentingChain(rsas, R); qff := RepresentingQuantifierFreeFormula(rsas); PP := PositiveInequalities(rsas, R); JSFH We compute the sample points of the connected components of the semi-algebraic sets described by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkcWZmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjk= JSFH R := PolynomialRing(allvars[4..-1]): sp := SamplePoints([S_0 > 0, E_0 > 0, C_0 > 0, km1 > 0, k1 > 0, C_0+E_0 <> 0, C_0+S_0 <> 0, E_0^2*k1^2-2*E_0*S_0*k1^2+S_0^2*k1^2+4*C_0*k1*km1+2*E_0*k1*km1+2*S_0*k1*km1+km1^2 <> 0], R); JSFH Display the coordinates of the first sample point. JSFH p1 := BoxValues(sp[1], R); JSFH Evaluate the regular chain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjcmNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGOQ==at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGOQ== rc1 := subs(p1, Equations(rc, R)); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkcmMxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjk=has only one real zero. R := PolynomialRing(allvars[1..3]); box := SamplePoints([op(rc1), S>0], R); Display(box, R); Display the coordinates of the second sample point. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= R := PolynomialRing(allvars[4..-1]); p2 := BoxValues(sp[2], R); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Evaluate the regular chain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjcmNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGOQ==at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjcDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGOQ== R := PolynomialRing(allvars[1..3]);rc2 := subs(p2, Equations(rc, R)); box := SamplePoints([op(rc2), S>0], R); Display(box, R); To conclude, the system has only one positive equilibrium if and only if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkcWZmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjk= holds.

JSFHJSFH 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A typical program on the left can be described by the semi-algebraic transition system on the right. JSFH Verifying the correctness of embedded software can be reduced to reachability computation of hybrid systems. For a class of hybrid system modelled by linear differential equations, the following paper [1] reduces the reachability problem to a quantifier elimination problem. [1] G. Lafferrierre, G. J. Pappas and S. Yovine: Symbolic reachability computation for families of linear vector fields. JSC, 2001.

JSFH The example is to compute the reachable set from (0,0) of a specific system and the problem is transformed into the following quantifier elimination problem. 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where restart;p1:=y1-2/3*a*(z-z^4);p2:=y2*z^2-1/2*a*(z^4-1); Decompose the system 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 with RealTriangularize. The output consists of five regular semi-algebraic systems. R := PolynomialRing([z, a, y1, y2]); rtd := RealTriangularize([p1=0, p2=0, a>=0,z-1>=0], R); Display(rtd, R); For the first four subsystems, we can immediately get conditions for the subsystems to have real solutions. We now consider the last one. r5 := rtd[5]; Display(r5, R); First we can immediately conclude that when 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above some real root of 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcjVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn has real solutions. However, the following computation shows that when 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5only has one real root. R := PolynomialRing([y1, y2]): p := 3*y1^5-6*y2*y1^4-63*y2^2*y1^3+192*y2^3*y1^2+112*y2^4*y1+16*y2^5; SamplePoints([p=0, y2>0], R); So after simplification, our final result is 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 print(); # input placeholder .

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlMtRiw2JVEld2l0aEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLlJlZ3VsYXJDaGFpbnNGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRlJGTUZqbkY5 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

We set S1 up first. C1:=[a > 0 , b > 0 , c > 0 , a < b + c , b < a + c , c < a + b, b^2 + a^2 - c^2 <= 0 ]: C2:=[a > 0 , b > 0 , c > 0 , a < b + c , b < a + c , c < a + b, c*(b^2 + a^2 - c^2)^2 < a*b^2*(2*a*c - (c^2 + a^2 - b^2))]: S1 is the disjuction of C1 and C2. S1:=[C1, C2];

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Then we set S2. S2 := [a-c<0, a > 0 , b > 0 , c > 0 , a < b + c , b < a + c , c < a + b];

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Compute regular semi-algebraic system representations dec1 (resp. dec2) for S1 (resp. S2) R := PolynomialRing([a,b,c]): dec1 := map(op, map(RealTriangularize, S1,R)); dec2:= RealTriangularize(S2, R);

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NyNJPnJlZ3VsYXJfc2VtaV9hbGdlYnJhaWNfc3lzdGVtRzYi

Compute the differences: S1 \134 S2 and S2 \134 S1 Difference(dec1,dec2,R);

NyI=

Difference(dec2,dec1,R);

NyI=

restart; R := PolynomialRing([y, x]);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

STBwb2x5bm9taWFsX3JpbmdHNiI=

f1 := sqrt(z^2-1)-sqrt(z+1)*sqrt(z-1); f2 := sqrt(1-z^2)-sqrt(1+z)*sqrt(1-z);

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LCYqJCwmIiIiRiUqJEkiekc2IiIiIyEiIiNGJUYpRiUqJiwmRidGJUYlRiVGKywmRiVGJUYnRipGK0Yq

f := f1; bc := RegularChains:-branchcuts([f]);

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NyU3JC9JInlHNiIiIiEySSJ4R0YmISIiNyRGJDJGKSIiIjckLywkKiZGKUYtRiVGLSIiI0YnMiwmKiRGKUYyRi0qJEYlRjJGKkYt

sp := CylindricalAlgebraicDecompose(bc, R, output=samplepoints); Display(sp, R); sp := map(BoxValues, sp, R):

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for p in sp do g := subs(z=rhs(p[2])+I*rhs(p[1]), f); g := simplify(g); print(g); end do:

LCQqJCIiJCMiIiIiIiNGJw==

IiIh

IiIh

KiZeIyIiIyIiIkYkI0YlRiQ=

IiIh

IiIh

IiIh

f := f2; bc := RegularChains:-branchcuts([f]);

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NyU3JC9JInlHNiIiIiEySSJ4R0YmISIiNyQvLCRGJUYqRicyLCRGKUYqRio3JC8sJComRikiIiJGJUY0ISIjRicyLCYqJEYpIiIjRioqJEYlRjlGNEYq

sp := CylindricalAlgebraicDecompose(bc, R, output=samplepoints); Display(sp, R); sp := map(BoxValues, sp, R):

NyRJJGJveEc2IkYj

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for p in sp do g := subs(z=rhs(p[2])+I*rhs(p[1]), f); g := simplify(g); print(g); end do:

IiIh

IiIh

restart;

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Start with the Cascad of Polymerisation. n := 4: RS := []: for i from 1 to n-1 do NewR1 := NewReaction(cat('P',i)+P1, cat('P',i+1), MassActionLaw(cat('k_',i)) ): NewR2 := NewReaction(cat('P',i+1), cat('P',i)+P1, MassActionLaw(cat('km_',i)) ): RS := [op(RS),NewR1,NewR2]: end do: RS;

NygtSSlSZWFjdGlvbkc2IjYmNyMtSSpTdWJzdHJhdGVHRiU2JEkjUDFHRiUiIiM3Iy1GKTYkSSNQMkdGJSIiIi1JLk1hc3NBY3Rpb25MYXdHRiU2I0kka18xR0YlSSZmYWxzZUclKnByb3RlY3RlZEctRiQ2JkYtRictRjM2I0kla21fMUdGJUY2LUYkNiY3JEYuLUYpNiRGK0YxNyMtRik2JEkjUDNHRiVGMS1GMzYjSSRrXzJHRiVGNi1GJDYmRkJGPy1GMzYjSSVrbV8yR0YlRjYtRiQ2JjckRkNGQDcjLUYpNiRJI1A0R0YlRjEtRjM2I0kka18zR0YlRjYtRiQ2JkZRRlAtRjM2I0kla21fM0dGJUY2

Claws := ConservationLaws(RS); Eqs := Equilibria(RS);

NyMsMkkjUDFHNiIjIiIiIiIlSSNQMkc2IiMiIiIiIiNJI1AzRzYiIyIiJCIiJUkjUDRHNiIiIiJJJVAxXzBHNiIjISIiIiIlSSVQMl8wRzYiIyEiIiIiI0klUDNfMEc2IiMhIiQiIiVJJVA0XzBHNiIhIiI=

NyUsLiomSSRrXzFHNiIiIiJJI1AxR0YmIiIjISIjKiZJJWttXzFHRiZGJ0kjUDJHRiZGJ0YpKihJJGtfMkdGJkYnRi1GJ0YoRichIiIqJkkla21fMkdGJkYnSSNQM0dGJkYnRicqKEkka18zR0YmRidGM0YnRihGJ0YwKiZJJWttXzNHRiZGJ0kjUDRHRiZGJ0YnLCpGLiMhIiRGKUYxIyIiJEYpRjQjRjBGKUY2I0YnRiksJkY0IyEiJUY9RjYjIiIlRj0=

F := [ op(Claws), op(Eqs) ]; N := []; vars := [P4, P3, P2, P1, k_1, k_2, k_3, km_1, km_2, km_3, P1_0, P2_0, P3_0, P4_0]: P := vars; H := []; R := PolynomialRing(vars);

NyYsMkkjUDFHNiIjIiIiIiIlSSNQMkdGJSNGJyIiI0kjUDNHRiUjIiIkRihJI1A0R0YlRidJJVAxXzBHRiUjISIiRihJJVAyXzBHRiUjRjJGK0klUDNfMEdGJSMhIiRGKEklUDRfMEdGJUYyLC4qJkkka18xR0YlRidGJEYrISIjKiZJJWttXzFHRiVGJ0YpRidGKyooSSRrXzJHRiVGJ0YpRidGJEYnRjIqJkkla21fMkdGJUYnRixGJ0YnKihJJGtfM0dGJUYnRixGJ0YkRidGMiomSSVrbV8zR0YlRidGL0YnRicsKkY/I0Y3RitGQSNGLkYrRkNGNEZFRiosJkZDIyEiJUYuRkUjRihGLg==

NyI=

NzBJI1A0RzYiSSNQM0dGJEkjUDJHRiRJI1AxR0YkSSRrXzFHRiRJJGtfMkdGJEkka18zR0YkSSVrbV8xR0YkSSVrbV8yR0YkSSVrbV8zR0YkSSVQMV8wR0YkSSVQMl8wR0YkSSVQM18wR0YkSSVQNF8wR0Yk

NyI=

STBwb2x5bm9taWFsX3JpbmdHNiI=

d := nops(vars) - n; rrc := RealRootClassification(numer(F), N, P, H, d, 1, R);

IiM1

NyQ3I0k7cmVndWxhcl9zZW1pX2FsZ2VicmFpY19zZXRHNiJJMmJvcmRlcl9wb2x5bm9taWFsR0Yl

Display(rrc[1][1], R); Display(rrc[2], R):

SSV0cnVlRyUqcHJvdGVjdGVkRw==

rrc := RealRootClassification(numer(F), N, P, H, d, 2..k, R);

NyQ3IkkyYm9yZGVyX3BvbHlub21pYWxHNiI=

Continue with the Allosteric Enzym restart;

SSV0cnVlRyUqcHJvdGVjdGVkRw==

R1 := NewReaction(E+S,C,MassActionLaw(k1)); R2 := NewReaction(C,E+S,MassActionLaw(k2)); R3 := NewReaction(E+C,FF,MassActionLaw(k3)); R4 := NewReaction(FF,E+C,MassActionLaw(k4)); RS := [R1,R2,R3,R4 ]:

LUkpUmVhY3Rpb25HNiI2JjckLUkqU3Vic3RyYXRlR0YkNiRJIkVHRiQiIiItRig2JEkiU0dGJEYrNyMtRig2JEkiQ0dGJEYrLUkuTWFzc0FjdGlvbkxhd0dGJDYjSSNrMUdGJEkmZmFsc2VHJSpwcm90ZWN0ZWRH

LUkpUmVhY3Rpb25HNiI2JjcjLUkqU3Vic3RyYXRlR0YkNiRJIkNHRiQiIiI3JC1GKDYkSSJFR0YkRistRig2JEkiU0dGJEYrLUkuTWFzc0FjdGlvbkxhd0dGJDYjSSNrMkdGJEkmZmFsc2VHJSpwcm90ZWN0ZWRH

LUkpUmVhY3Rpb25HNiI2JjckLUkqU3Vic3RyYXRlR0YkNiRJIkVHRiQiIiItRig2JEkiQ0dGJEYrNyMtRig2JEkjRkZHRiRGKy1JLk1hc3NBY3Rpb25MYXdHRiQ2I0kjazNHRiRJJmZhbHNlRyUqcHJvdGVjdGVkRw==

LUkpUmVhY3Rpb25HNiI2JjcjLUkqU3Vic3RyYXRlR0YkNiRJI0ZGR0YkIiIiNyQtRig2JEkiRUdGJEYrLUYoNiRJIkNHRiRGKy1JLk1hc3NBY3Rpb25MYXdHRiQ2I0kjazRHRiRJJmZhbHNlRyUqcHJvdGVjdGVkRw==

Claws := ConservationLaws(RS); Eqs := Equilibria(RS);

NyQsLkkiQ0c2IiMiIiIiIiNJIkVHRiUjISIiRihJIlNHRiVGJ0kkQ18wR0YlRipJJEVfMEdGJUYmSSRTXzBHRiVGKywuRiRGJkYpRiZJI0ZGR0YlRidGLUYqRi5GKkklRkZfMEdGJUYr

NyQsKiooSSNrMUc2IiIiIkkiRUdGJkYnSSJTR0YmRidGJyomSSNrMkdGJkYnSSJDR0YmRichIiIqKEkjazNHRiZGJ0YoRidGLEYnRi0qJkkjazRHRiZGJ0kjRkZHRiZGJ0YnLCZGLiEiI0YwIiIj

F := [ op(Claws), op(Eqs) ]; N := []; vars := [FF, E, S, C, k2, k1, k3, k4, C_0, E_0, FF_0, S_0]; P := vars; H := []; R := PolynomialRing(vars);

NyYsLkkiQ0c2IiMiIiIiIiNJIkVHRiUjISIiRihJIlNHRiVGJ0kkQ18wR0YlRipJJEVfMEdGJUYmSSRTXzBHRiVGKywuRiRGJkYpRiZJI0ZGR0YlRidGLUYqRi5GKkklRkZfMEdGJUYrLCoqKEkjazFHRiVGJ0YpRidGLEYnRicqJkkjazJHRiVGJ0YkRidGKyooSSNrM0dGJUYnRilGJ0YkRidGKyomSSNrNEdGJUYnRjFGJ0YnLCZGOCEiI0Y6Rig=

NyI=

Ny5JI0ZGRzYiSSJFR0YkSSJTR0YkSSJDR0YkSSNrMkdGJEkjazFHRiRJI2szR0YkSSNrNEdGJEkkQ18wR0YkSSRFXzBHRiRJJUZGXzBHRiRJJFNfMEdGJA==

Ny5JI0ZGRzYiSSJFR0YkSSJTR0YkSSJDR0YkSSNrMkdGJEkjazFHRiRJI2szR0YkSSNrNEdGJEkkQ18wR0YkSSRFXzBHRiRJJUZGXzBHRiRJJFNfMEdGJA==

NyI=

STBwb2x5bm9taWFsX3JpbmdHNiI=

d := nops(vars) - nops(F); rrc := RealRootClassification(F, N, P, H, d, 2..k, R); rrc := RealRootClassification(F, N, P, H, d, 1, R);

IiIp

NyQ3IkkyYm9yZGVyX3BvbHlub21pYWxHNiI=

NyQ3I0k7cmVndWxhcl9zZW1pX2FsZ2VicmFpY19zZXRHNiJJMmJvcmRlcl9wb2x5bm9taWFsR0Yl

Display(rrc[2], R); Display(rrc[1], R);

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

NyNJJXRydWVHJSpwcm90ZWN0ZWRH

vars := [FF, k2, E, S, k1, C, k3, k4, C_0, E_0, FF_0, S_0]; R := PolynomialRing(vars); dec := RealTriangularize(F, N, P, H, R);

Ny5JI0ZGRzYiSSNrMkdGJEkiRUdGJEkiU0dGJEkjazFHRiRJIkNHRiRJI2szR0YkSSNrNEdGJEkkQ18wR0YkSSRFXzBHRiRJJUZGXzBHRiRJJFNfMEdGJA==

STBwb2x5bm9taWFsX3JpbmdHNiI=

NyNJPnJlZ3VsYXJfc2VtaV9hbGdlYnJhaWNfc3lzdGVtRzYi

JSFH The problem is transformed into the following quantifier elimination problem. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkobWZlbmNlZEdGJDYkLUYjNictSSNtb0dGJDYtUSkmRXhpc3RzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy1GMTYtUSJ+RidGNEY3RjpGPEY+RkBGQkZERkYvRkpGSC1JI21pR0YkNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNVEnaXRhbGljRicvJStleGVjdXRhYmxlR0Y5RjRGNC1GLDYkLUYjNidGMEZMLUZRNiVRInpGJ0ZURldGWUY0RjQtRiw2JC1GIzY3LUZRNiVRI3AxRidGVEZXLUYxNi1RIj1GJ0Y0RjdGOkY8Rj5GQEZCRkQvRkdGS0ZJLUkjbW5HRiQ2JFEiMEYnRjRGTC1GMTYtUSYmYW5kO0YnRjRGN0Y6L0Y9RlZGPkZARkJGRC9GR1EsMC4yMjIyMjIyZW1GJy9GSkZgcEZMLUZRNiVRI3AyRidGVEZXRmNvRmdvRkxGW3BGUC1GMTYtUS8mR3JlYXRlckVxdWFsO0YnRjRGN0Y6RjxGPkZARkJGREZmb0ZJRmdvRkwtRjE2LVEoJndlZGdlO0YnRjRGN0Y6RjxGPkZARkJGRC9GR1EsMC4xNjY2NjY3ZW1GJy9GSkZccUZpbkZlcC1GaG82JFEiMUYnRjRGWUY0RjRGTEZZRjQ=where restart;p1:=y1-2/3*a*(z-z^4);p2:=y2*z^2-1/2*a*(z^4-1);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LCZJI3kxRzYiIiIiKiZJImFHRiRGJSwmSSJ6R0YkRiUqJEYpIiIlISIiRiUjISIjIiIk

LCYqJkkjeTJHNiIiIiJJInpHRiUiIiNGJiomSSJhR0YlRiYsJiokRiciIiVGJiEiIkYmRiYjRi5GKA==

Decompose the system 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 with RealTriangularize. The output consists of five regular semi-algebraic systems. R := PolynomialRing([z, a, y1, y2]); rtd := RealTriangularize([p1=0, p2=0, a>=0,z-1>=0], R); Display(rtd, R):

STBwb2x5bm9taWFsX3JpbmdHNiI=

NydJPnJlZ3VsYXJfc2VtaV9hbGdlYnJhaWNfc3lzdGVtRzYiRiNGI0YjRiM=

r1 := rtd[1]; Display(r1, R);

ST5yZWd1bGFyX3NlbWlfYWxnZWJyYWljX3N5c3RlbUc2Ig==

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

r2 := rtd[2]; Display(r2, R);

ST5yZWd1bGFyX3NlbWlfYWxnZWJyYWljX3N5c3RlbUc2Ig==

LUkqUElFQ0VXSVNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNig3JC1JJW1yb3dHNiRGJS9JK21vZHVsZW5hbWVHRidJLFR5cGVzZXR0aW5nR0YkNiUtRis2LS1GKzYlLUkobWZlbmNlZEdGLDYjLUYrNiotSSNtb0dGLDYjUSomdW1pbnVzMDtGJy1GKzYlLUkjbW5HRiw2I1EiOEYnLUY7NiNRMSZJbnZpc2libGVUaW1lcztGJy1GKzYlLUklbXN1cEdGLDYkLUkjbWlHRiw2I1EiYUYnLUZBNiNRIjNGJ0ZELUZNNiNRI3kyRictRjs2I1EoJm1pbnVzO0YnLUYrNiUtRkE2I1EjMTJGJ0ZELUYrNictRko2JEZMLUZBNiNRIjJGJ0ZELUZNNiNRI3kxRidGREZTRlYtRis2JS1GQTYjUSMxNkYnRkQtRis2JUZMRkQtRko2JEZTRlAtRjs2I1EnJnBsdXM7RictRis2JUZcb0ZELUZKNiRGTC1GQTYjUSI0RidGRC1GTTYjUSJ6RidGVkZecEZbcC1GKzYlLUY2NiMtRis2JS1GKzYlRmJwRkRGU0ZWLUYrNiVGUEZERl9vRkRGSUZbcC1GKzYpRmVuRkRGam5GREZfb0ZERlNGW3AtRis2KS1GQTYjUSI5RidGREZMRkQtRko2JEZfb0Zcb0ZERlNGW3AtRis2Jy1GQTYjUSMyNEYnRkRGX29GREZpby1GOzYjUSkmZXF1YWxzO0YnLUZBNiNRIjBGJ0khR0YnNyQtRis2JS1GKzYzLUYrNiUtRjY2Iy1GKzYrLUYrNiUtRkE2I1FQNTk5NDIwMTk0OTYyNDgxNzEyOTk4NDM2OTc2OTY0MTg3MzU3MDY2MjM3ODM5MzZGJ0ZELUYrNiVGX29GREZpb0ZbcC1GKzYlLUZBNiNRUDkzNDM5OTcxNjU3NDk5MzgzNjgwOTc0NjA5NDczNjE1MTcyMjY4MTIyMzU0MTc2RidGRC1GKzYlRmlxRkQtRko2JEZTRlxvRlYtRis2JS1GQTYjUVA1NDE3NzUyMjU4ODA4NjkyNDU5MTU3NzkyNjE5MzcwODMyMzAzMzkxMTAyMzIxNkYnRkQtRis2JS1GSjYkRl9vRlBGREZTRltwLUYrNiUtRkE2I1FPODgyNTI5MzMyNDg2OTU2ODU2Nzg0MzEyMTg4MTIyNzc1MjM0MjkzMjAwMzA3MkYnRkQtRko2JEZTRmJwRltwLUYrNiUtRkE2I1FPNzk1NzMzMTIyODMzNjA3MTYwNTU3NTAwMDk2MTAxMDA2MTkzMTUwNDMyNTg3MkYnRkQtRko2JEZfb0ZicEZERmBwRltwLUYrNiUtRjY2Iy1GKzYsRjotRis2JS1GQTYjUVE2NjY3NjgyMjI3NjI2NTM0OTQyMzUxODUxODA0NTYzODc0Nzg1NDA1NDM1OTUyMDBGJ0ZELUYrNiVGYHVGREZfb0ZWLUYrNiUtRkE2I1FQODg1MDc3OTYwOTIxMzQwMDk2MTE1MjAxNzU0ODQ1MDM0MTQxMzcyNDExNDg4MjBGJ0ZELUYrNiVGU0ZERmd1RlYtRis2JS1GQTYjUVA5ODE2ODg0OTI2ODgzOTU1OTE3ODU1MDc2ODg5MTAyMTg5MDA5MjE1MTczOTI2NEYnRkQtRko2JEZTLUZBNiNRIjVGJ0ZbcC1GKzYlLUZBNiNRUTYwMjY3MzIzMzQ0ODcwNjQwNTUyNTI2NDAwMTI1NzYwNTg3MzYzNzMxNzk4NDY0OEYnRkQtRis2JUZgdEZERml0RlYtRis2JS1GQTYjUVIxMDM5Mzc1MDExODE2NTA1MTgyNTIwNTY5MTY0NjMzMjg4MjAxMzQxNDIxMTMxOTY4RidGRC1GKzYlRmlvRkRGaXFGREZJRltwLUYrNiUtRjY2Iy1GKzYrLUYrNiUtRkE2I1FSMzQ5MDk2NzAzMDU1OTIzNTA5Njg2MjQ3MDUyOTMwNDY2Njg4NDgzODM4MzQ4Njk3NkYnRkQtRis2JUZfb0ZERmJ3RltwLUYrNiUtRkE2I1FRNTEzOTc5NDYzNTc0OTc5MDExMzIxMzEwNzcwNDU3NzgzMDExMjU4MTk5ODkzNDU2RidGRC1GSjYkRlMtRkE2I1EiNkYnRltwLUYrNiUtRkE2I1FSNTQ0MTc1NTk0Nzk5MjMxMDYxMzQwMDM5NTY1OTE4NzU2NzIwMTU2MDUyNTQyOTYwMEYnRkQtRis2JUZpcUZERmB1RltwLUYrNiUtRkE2I1FRNDYzNDYzMjYxMjUwODcyMDM4MTc1NDI1MzI0NDc5MzExNDY4OTI2NDA3NjUwNjYzRidGRC1GKzYlRmd1RkRGYHRGVi1GKzYlLUZBNiNRUjMxNTU0OTg1OTE0NTk0MTg2NjIyODMxNzAxMzQzMzM1NjE2Mzk1OTA1NzM5NzU5MDdGJ0ZELUYrNiVGaXRGREZpb0ZERmpuRltwLUYrNiUtRjY2Iy1GKzYsRjotRis2JS1GQTYjUVI3NTM2Nzk2ODI4Nzg1Mjg4MzA5NDM5OTQ3ODUxNDY3MjA1Nzc4MTI0NTY5MjkwNzUyRidGRC1GKzYlRmlxRkRGYndGVi1GKzYlLUZBNiNRUjQ4MzQ4OTg3MDkzMzgzNDU2NDk4NTc5OTgxOTk2NzIyOTg4NDI1NDI1ODIyMzE1NTJGJ0ZELUYrNiVGX29GREZneUZWLUYrNiUtRkE2I1FRNjQxNzg5MjIzODY3ODkxNDAxOTYwMzg3MzY0NzA2MzY1NzE4MDg3ODAzMjk1NDQwRidGRC1GKzYlRmd1RkRGaW9GVi1GKzYlLUZBNiNRUTcxMTg0NTMwNDY2MTAxNjkzMTUyMDQxNjI5ODkyODc2MDI3NTY1NzcxOTE3NTQyNEYnRkQtRko2JEZTLUZBNiNRIjdGJ0ZbcC1GKzYlLUZBNiNRUjQzNzAwMzgxNTg3NzU4NTEwMzAyNDIwODk0MjM4NzY0ODY1MTIyNTM1NTE2OTM5MzZGJ0ZELUYrNiVGaXRGREZgdUZERkxGVi1GKzYnLUZBNiNRUjE1NDg1MTc1MDUyNTgzMDYxNjk5NTkyNDgzMzI2ODQ5MzUyNDc0MDg3NzE0MjEyNDhGJ0ZERml0RkRGYndGW3AtRis2Jy1GQTYjUVIyNjcwNDg3NzI5NjcyMDI5NTcxMTc0ODU3MjI2ODQ4Mjc4OTQ2Mzc1OTE4OTI1ODI0RidGREZpcUZERmd5RltwLUYrNictRkE2I1FSMTcxMzE1NTA0NjY5MDQ1MjEzODQ5ODYzMDM4NDQ3NjMxODgxNzM0NTU0NjMxNzgyNEYnRkRGX29GREZhXWxGW3AtRis2Jy1GQTYjUVEyMjc0NDA0ODEzODkxODI2MDE4MjQ0Mzg1OTU2NTUyNDY2NzczNzUxMTE1NTQ2MjRGJ0ZERmd1RkRGYHVGW3AtRis2JS1GQTYjUVEyNTIyMjk4NzY2MTkzOTg4Nzk0MjY5MjA3MDU2MDQ2NzkyMjM0MzA4NzY0MTI5MjhGJ0ZELUZKNiRGU0ZARmByRmNyRmZyNyQtRis2JS1GKzYtLUYrNiVGUEZELUZKNiRGX29GZHdGVi1GKzYlRml5RkRGW3dGVi1GKzYlLUZBNiNRIzYzRidGREZceEZbcC1GKzYlLUZBNiNRJDE5MkYnRkRGY3hGW3AtRis2JS1GQTYjUSQxMTJGJ0ZERmR2RltwLUYrNiVGZG9GREZid0ZgckZjckZmcjckLUYrNiUtRis2JUZlcEZWLUZBNiNRIjFGJy1GOzYjUSUmZ3Q7RidGY3JGZnI3JC1GKzYlRkxGXGJsRmNyRmZyNyQtRis2JUZTRlxibEZjckZmcg==

R12 := PolynomialRing([y1, y2]); p := 3*y1^5-6*y2*y1^4-63*y2^2*y1^3+192*y2^3*y1^2+112*y2^4*y1+16*y2^5; SamplePoints([p=0, y2>0], R12);

STBwb2x5bm9taWFsX3JpbmdHNiI=

LC4qJEkjeTFHNiIiIiYiIiQqJkkjeTJHRiUiIiJGJCIiJSEiJyomRikiIiNGJEYnISNqKiZGKUYnRiRGLiIkIz4qJkYpRitGJEYqIiQ3IiokRilGJiIjOw==

NyNJJGJveEc2Ig==

r3 := rtd[3]; Display(r3, R); r4 := rtd[4]; Display(r4, R); r4 := rtd[4]; Display(r4, R);

ST5yZWd1bGFyX3NlbWlfYWxnZWJyYWljX3N5c3RlbUc2Ig==

LUkqUElFQ0VXSVNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JC1JJW1yb3dHNiRGJS9JK21vZHVsZW5hbWVHRidJLFR5cGVzZXR0aW5nR0YkNiUtRis2JS1JI21pR0YsNiNRInpGJy1JI21vR0YsNiNRKCZtaW51cztGJy1JI21uR0YsNiNRIjFGJy1GODYjUSkmZXF1YWxzO0YnLUY8NiNRIjBGJ0khR0YnNyQtRis2JS1GNDYjUSJhRidGP0ZCRkU3JC1GKzYlLUY0NiNRI3kxRidGP0ZCRkU3JC1GKzYlLUY0NiNRI3kyRidGP0ZCRkU=

ST5yZWd1bGFyX3NlbWlfYWxnZWJyYWljX3N5c3RlbUc2Ig==

LUkqUElFQ0VXSVNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JC1JJW1yb3dHNiRGJS9JK21vZHVsZW5hbWVHRidJLFR5cGVzZXR0aW5nR0YkNiUtRis2JS1JI21pR0YsNiNRInpGJy1JI21vR0YsNiNRKCZtaW51cztGJy1JI21uR0YsNiNRIjFGJy1GODYjUSkmZXF1YWxzO0YnLUY8NiNRIjBGJ0khR0YnNyQtRis2JS1GNDYjUSN5MUYnRj9GQkZFNyQtRis2JS1GNDYjUSN5MkYnRj9GQkZFNyQtRis2JS1GNDYjUSJhRictRjg2I1ElJmd0O0YnRkJGRQ==

ST5yZWd1bGFyX3NlbWlfYWxnZWJyYWljX3N5c3RlbUc2Ig==

LUkqUElFQ0VXSVNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JC1JJW1yb3dHNiRGJS9JK21vZHVsZW5hbWVHRidJLFR5cGVzZXR0aW5nR0YkNiUtRis2JS1JI21pR0YsNiNRInpGJy1JI21vR0YsNiNRKCZtaW51cztGJy1JI21uR0YsNiNRIjFGJy1GODYjUSkmZXF1YWxzO0YnLUY8NiNRIjBGJ0khR0YnNyQtRis2JS1GNDYjUSN5MUYnRj9GQkZFNyQtRis2JS1GNDYjUSN5MkYnRj9GQkZFNyQtRis2JS1GNDYjUSJhRictRjg2I1ElJmd0O0YnRkJGRQ==

So after simplification, our final result is 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 print(); # input placeholder .

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZmby1JI21pR0YkNiVRI2YxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RKiZjb2xvbmVxO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZTRjUtRiw2JVEjazFGJ0YvRjItRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmZuLUYsNiVRI2syRidGL0YyLUY2Ni1RIipGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GX28tRiw2JVEieEYnRi9GMkZYRmFvRltvLUYsNiVRImFGJ0YvRjJGW28tSShtZmVuY2VkR0YkNiQtRiM2Ky1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGZW5GZ24tRiw2JVEiYkYnRi9GMkZbby1GLDYlUSJ5RidGL0YyLUY2Ni1RIl5GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GXXEtRiw2JVEibkYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUY2Ni1RIi9GJ0Y5RjtGPi9GQUYxRkJGREZGRkhGXm9GYG8tRmhvNiQtRiM2K0ZccEZgcC1GLDYlUSJjRidGL0YyRltvRmZwRmlwRl9xRmJxRjlGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRkpGVC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZMLyUmZGVwdGhHRmhyLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZkcjYmRmZyRmlyRltzL0Zec1ElYXV0b0YnLUYsNiVRI2YyRidGL0YyRjVGT0Y1RmFvRltvRmRvRltvRmdvRmRxRmhxRlgtRiw2JVEjazRGJ0YvRjJGW29GZnBGX3JGY3JGYHNGK0Y1Rk9GNS1GLDYlUSVzdWJzRidGL0YyLUZobzYkLUYjNkdGY3AtRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZSRlQtRl1wNiRRIjJGJ0Y5LUY2Ni1RIixGJ0Y5RjtGYnJGQEZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRidGNUZcckZhdEZccEZkcS1GXXA2JFEjMjBGJ0Y5Rmd0RjVGX3FGYXQtRl1wNiRRIjRGJ0Y5Rmd0RjVGZG9GYXRGXHBGZHEtRl1wNiRRIzEwRidGOUZndEY1RmdzRmF0LUZdcDYkUSM1MEYnRjlGZ3RGNUZVRmF0LUZdcDYkUSQ4MDBGJ0Y5Rmd0RjVGK0ZicUY5RjlGX3JGY3JGYHNGZHNGNUZPRjVGanMtRmhvNiQtRiM2R0ZjcEZhdEZkdEZndEY1RlxyRmF0RlxwRmRxRlx1Rmd0RjVGX3FGYXRGX3VGZ3RGNUZkb0ZhdEZccEZkcUZidUZndEY1RmdzRmF0RmV1Rmd0RjVGVUZhdEZodUZndEY1RmRzRmJxRjlGOUZfckZjckZgcy1GLDYlUSNwMUYnRi9GMkY1Rk9GNS1GLDYlUSZudW1lckYnRi9GMi1GaG82JC1GIzYlRitGYnFGOUY5Rl9yRmNyRmBzLUYsNiVRI3AyRidGL0YyRjVGT0Y1RmJ2LUZobzYkLUYjNiVGZHNGYnFGOUY5Rl9yRmJxRjk= 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 JSFH

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