Reduce[(x^2)+(y^2)==(z^2)&&(Subscript[∃, t])((Subscript[∀, u])t x≠u y z+7||(x^2) t==2 z+1),{x,y,z}]

Resolve[(x^2)+(y^2)==(z^2)&&(Subscript[∃, t])((Subscript[∀, u])t x≠u y z+7||(x^2) t==2 z+1)]

FindInstance[(x^2)+(y^2)==(z^2)&&(Subscript[∃, t])((Subscript[∀, u])t x≠u y z+7||(x^2) t==2 z+1),{x,y,z}]

(RefLink[GroebnerBasis, paclet:ref/GroebnerBasis]) options used in the solving of complex polynomial systems.

CoefficientDomain

Available settings for CoefficientDomain.

Method

Transforming Gröbner bases using the Gröbner walk algorithm.

MonomialOrder

Monomial orders.

Gröbner basis in elimination order.

(RefLink[SetSystemOptions, paclet:ref/SetSystemOptions])["GroebnerBasisOptions"->{"EliminateFromGroebnerBasis"->(RefLink[False, paclet:ref/False])}].

System option EliminateFromGroebnerBasis.

GroebnerBasis[{(Subscript[x, 1]^2)+(Subscript[x, 2]^2)-(Subscript[x, 1])(Subscript[x, 2])y-1,(Subscript[x, 1]^2)+(Subscript[x, 2]^2)+(Subscript[x, 1])(Subscript[x, 2])y+1},{(Subscript[x, 1]),(Subscript[x, 2])},{y},MonomialOrder->EliminationOrder]

SetSystemOptions["GroebnerBasisOptions"->{"EliminateFromGroebnerBasis"->False}];
GroebnerBasis[{(Subscript[x, 1]^2)+(Subscript[x, 2]^2)-(Subscript[x, 1])(Subscript[x, 2])y-1,(Subscript[x, 1]^2)+(Subscript[x, 2]^2)+(Subscript[x, 1])(Subscript[x, 2])y+1},{(Subscript[x, 1]),(Subscript[x, 2])},{y},MonomialOrder->EliminationOrder]

SetSystemOptions["GroebnerBasisOptions"->{"EliminateFromGroebnerBasis"->True}];

Resolve[(Subscript[∃, y])((Subscript[x, 1]^2)+(Subscript[x, 2]^2)-(Subscript[x, 1])(Subscript[x, 2])y==1∧(Subscript[x, 1]^2)+(Subscript[x, 2]^2)+(Subscript[x, 1])(Subscript[x, 2])y+1==0)]

Reduce[(Subscript[∃, {a,b,c,x,y}])(a (x^2)+b x+c==0&&a (y^2)+b y+c==0&&x≠y&&(b^2)-4 a c==0&&a≠0)]

GroebnerBasis[{(x^2)+(y^2)-2,x-y,1-(x+1) z},{x,y,z}]

GroebnerBasis[{(x^2)+(y^2)-2,x-y,1-((x^2)-1) z},{x,y,z}]

Resolve[(Subscript[∃, {x,y}])(a (x^2)+b x+c==0&&a (y^2)+b y+c==0&&x≠y)]

SetSystemOptions["GroebnerBasisOptions"->{"EliminateFromGroebnerBasis"->False}];
GroebnerBasis[{(Subscript[x, 1]^2)+(Subscript[x, 2]^2)+(y^2)-1,(Subscript[x, 1])+(Subscript[x, 2])-y},{(Subscript[x, 1]),(Subscript[x, 2])},{y},MonomialOrder->EliminationOrder]

SetSystemOptions["GroebnerBasisOptions"->{"EliminateFromGroebnerBasis"->True}];

GroebnerBasis[{((((Subscript[x, 2])-(Subscript[x, 3])))^2) ((Subscript[x, 2])-2 (Subscript[x, 3])),((Subscript[x, 2])-(Subscript[x, 3])) (Subscript[x, 1]),(Subscript[x, 1]^2)-(Subscript[x, 1])},{(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])}]

GroebnerBasis[{(Subscript[x, 2]^3)-4 (Subscript[x, 2]^2) (Subscript[x, 3])+5 (Subscript[x, 2]) (Subscript[x, 3]^2)-2 (Subscript[x, 3]^3),(Subscript[x, 1]) (Subscript[x, 2])-(Subscript[x, 1]) (Subscript[x, 3]),-(Subscript[x, 1])+(Subscript[x, 1]^2),(Subscript[x, 2])-(Subscript[x, 3]) },{(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])}]

Options of (RefLink[Reduce, paclet:ref/Reduce]) and (RefLink[Resolve, paclet:ref/Resolve]) affecting the behavior of complex polynomial systems.

Options of (RefLink[Reduce, paclet:ref/Reduce]), (RefLink[Resolve, paclet:ref/Resolve]), and (RefLink[FindInstance, paclet:ref/FindInstance]) affecting the behavior of complex polynomial systems.

Backsubstitution

Reduce[(x^2)+(y^2)==1&&(x^5)-3 x+7==0,{x,y}]

Reduce[(x^2)+(y^2)==1&&(x^5)-3 x+7==0,{x,y},Backsubstitution->True]

Cubics and Quartics

Reduce[(x^3)-3 x+7==0, x]

Reduce[(x^3)-3 x+7==0, x, Cubics->True]

WorkingPrecision

Reduce[x^2+y^2==1&&x^3-y^3==2, {x,y}, WorkingPrecision->20]

(RefLink[SetSystemOptions, paclet:ref/SetSystemOptions])["ReduceOptions"->{"option name"->value}].

SetSystemOptions["ReduceOptions"->{"FinitePrecisionGB"->True}];

"FinitePrecisionGB"/.("ReduceOptions"/.SystemOptions[])

SetSystemOptions["ReduceOptions"->{"FinitePrecisionGB"->False}];

ReduceOptions group options that affect the behavior of (RefLink[Reduce, paclet:ref/Reduce]), (RefLink[Resolve, paclet:ref/Resolve]), and (RefLink[FindInstance, paclet:ref/FindInstance]) for complex polynomial systems.

AlgebraicNumberOutput

Reduce[(y^3)-3 (y^2)+(x^3)+4==0&&(y^2)+y-(x^2)-3 x-11==0,{x,y},Backsubstitution->True]

SetSystemOptions["ReduceOptions"->"AlgebraicNumberOutput"->False];
Reduce[(y^3)-3 (y^2)+(x^3)+4==0&&(y^2)+y-(x^2)-3 x-11==0,{x,y},Backsubstitution->True]

SetSystemOptions["ReduceOptions"->"AlgebraicNumberOutput"->True];

FinitePrecisionGB

SeedRandom[123];
f=(∑_(i=0)^3)(∑_(j=0)^4)RandomInteger[{-(10^100),(10^100)}] (x^i) (y^j);g=(∑_(i=0)^4)(∑_(j=0)^3)RandomInteger[{-(10^100),(10^100)}] (x^i) (y^j);
Timing[(Subscript[a, 1])=Reduce[f==0&&g==0,{x,y},WorkingPrecision->200,Backsubstitution->True];]

SetSystemOptions["ReduceOptions"->{"FinitePrecisionGB"->True}];
Timing[(Subscript[a, 2])=Reduce[f==0&&g==0,{x,y},WorkingPrecision->200,Backsubstitution->True];]

Precision/@{(Subscript[a, 1]),(Subscript[a, 2])}

Sort[{x,y}/.{ToRules[(Subscript[a, 1])]}]-Sort[{x,y}/.{ToRules[(Subscript[a, 2])]}]

SetSystemOptions["ReduceOptions"->{"FinitePrecisionGB"->False}];

ReorderVariables

Reduce[(z^3)+3 z-2 y+1==x&&(z^2)-7==y,{x,y,z}]

SetSystemOptions["ReduceOptions"->{"ReorderVariables"->True}];Reduce[(z^3)+3 z-2 y+1==x&&(z^2)-7==y,{x,y,z}]

SetSystemOptions["ReduceOptions"->{"ReorderVariables"->Automatic}];

Solve[x==(y^3)+7 y-1&&(y^5)-y==2,{x,y}]

SetSystemOptions["ReduceOptions"->"ReorderVariables"->False];
Solve[x==(y^3)+7 y-1&&(y^5)-y==2,{x,y}]

SetSystemOptions["ReduceOptions"->"ReorderVariables"->Automatic];

UseNestedRoots

FindInstance[(x^5)+2x+1==0&&(y^5)+3 (x^2) y+1==0,{x,y}]

SetSystemOptions["ReduceOptions"->"UseNestedRoots"->False];
FindInstance[(x^5)+2x+1==0&&(y^5)+3 (x^2) y+1==0,{x,y}]

SetSystemOptions["ReduceOptions"->"UseNestedRoots"->Automatic];
FindInstance[(x^5)+2x+1==0&&(y^5)+3 (x^4) y+1==0&&(z^5)+3 (y^4) z+1==0,{x,y,z}]//Timing

SetSystemOptions["ReduceOptions"->"UseNestedRoots"->False];
FindInstance[(x^5)+2x+1==0&&(y^5)+3 (x^4) y+1==0&&(z^5)+3 (y^4) z+1==0,{x,y,z}]//Timing

SetSystemOptions["ReduceOptions"->"UseNestedRoots"->True];
Solve[(x^3)+x (y^2)-x-1==0&&(y^3)+3 (x^2) y+1==0,{x,y}]

SetSystemOptions["ReduceOptions"->"UseNestedRoots"->Automatic];
Solve[(x^3)+x (y^2)-x-1==0&&(y^3)+3 (x^2) y+1==0,{x,y}]