Base Algebra

Projection := proc ( sys  ::list(algebraic),    vars1  ::list({name,name=algebraic}),    vars2  ::list(name)  :=  [op(indets(sys,name) minus {op(vars1)} minus indets(sys,constant))], char  ::integer  :=  0, {      output  ::identical(list,listlist)  :=  'list', allvariables  ::truefalse  :=  false, keeporder  ::truefalse  :=  false, nofactor  ::truefalse  :=  false }      )

Project on variables or expressions in a polynomial system.

Parameters

Returns

With output=list, a list of polynomials: the generators of the elimination ideal.  With output=listlist, a list of components of the elimination ideal (multiplicities may have been lost); each component is represented by a list of polynomials.

Example

> Projection ( [x+y^2,y-x], [z=x+y] );
< [z^2+2*z]

TrigonometricBasis := proc( sys  ::list(algebraic),    vars1  ::list(name),    vars2  ::list(name)  :=  [op(indets(sys,name) minus {op(vars1)})], char  ::integer  :=  0 )

Computes a Gröbner basis of a trigonometric system.

Parameters

Example

> Elimination ( [x+y^2,y], [y], [x] );
< [x]

Elimination := proc ( sys  ::list(algebraic),    vars1  ::list(name),    vars2  ::list(name)  :=  [op(indets(sys,name) minus {op(vars1)})], char  ::integer  :=  0, {      allvariables  ::truefalse  :=  false, keeporder  ::truefalse  :=  false, direct  ::truefalse  :=  false, incremental  ::truefalse  :=  false, generic  ::truefalse  :=  false }           )

Eliminates variables in a polynomial system.

Parameters

Returns

A list of polynomials: the generators of the elimination ideal.

Example

> Elimination ( [x+y^2,y], [y], [x] );
< [x]

Intersection := proc ( sys1  ::list(algebraic),    sys2  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys1,sys2],name))], char  ::integer  :=  0 )

Intersects two ideals.

Parameters

Returns

A list of polynomials: the generators of the intersection ideal.

Example

> Intersection ( [x,z], [y] );
< [x*y,z*y]

Division := proc ( sys1  ::list(algebraic),    sys2  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys1,sys2],name))], char  ::integer  :=  0 )

Division of an ideal by another.

Parameters

Returns

A list of polynomials: the generators of the division ideal.

Example

> Division ( [x^3,z], [x] );
< [x^2,z]

NormalForm := proc ( polys  ::list(algebraic),    sys  ::list(algebraic),    vars1  ::list(name)  :=  NULL, vars2  ::list(name)  :=  [], char  ::integer  :=  0 )

Reduces polynomials w.r.t a Groebner basis.

Parameters

Returns

A list of polynomials: the normal forms of the polynomials in polys up to a scalar factor.

Example

> NormalForm ( [x^2+1,x+y], [x^2,y], [x,y] );
< [1,x]

InRadical := proc ( p  ::algebraic,    sys  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys,p],name))], char  ::integer  :=  0 )

Test the membership to the radical of an ideal.

Parameters

Returns

A boolean: true if p is in the radical of sys, false otherwise.

Example

> InRadical ( x, [x^2,y] );
< true

Dimension := proc ( sys  ::list(algebraic),    vars  ::list(name)  :=  [op (indets (sys,name))], char  ::integer  :=  0 )

Computes the dimension of a system.

Parameters

Returns

An integer: the dimension of the system.

Example

> Dimension ( [x^2,cos(y)-0.5] );
< 0

SaturationIdeal := proc ( sys1  ::list(algebraic),    sys2  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys1,sys2],name))], char  ::integer  :=  0, {      alternative  ::truefalse  :=  false }           )

Saturate an ideal by another ideal.

Parameters

Returns

A list of polynomials: the generators of sys1 saturated by sys2.

Example

> SaturationIdeal ( [x*y*z], [z*x,z*y] )
< [x*y]

SaturationTest := proc ( sys1  ::list(algebraic),    sys2  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys1,sys2],name))], char  ::integer  :=  0 )

Saturate an ideal by another ideal (new method).

Parameters

Returns

A list of polynomials: the generators of sys1 saturated by sys2.

Example

> SaturationTest ( [x*y*z], [z*x,z*y] )
< [x*y]

Algorithm

Saturation5 := proc ( sys1  ::list(algebraic),    sys2  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys1,sys2],name))], char  ::integer  :=  0, outpol  ::algebraic  :=  1 )

Saturate an ideal by another ideal (new method).

Parameters

Returns

A list of polynomials: the generators of sys1 saturated by sys2.

Example

> SaturationTest2 ( [x*y*z], [z*x,z*y] )
< [x*y]

SaturationDirect := proc ( sys1  ::list(algebraic),    sys2  ::list(algebraic),    vars  ::list(name)  :=  [op (indets ([sys1,sys2],name))], char  ::integer  :=  0 )

Saturate an ideal by another ideal (direct method).

Parameters

Returns

A list of polynomials: the generators of sys1 saturated by sys2.

Example

> SaturationDirect ( [x*y*z], [z*x,z*y] )
< [x*y]

IteratedJacobian := proc ( sys  ::list(algebraic),    vars  ::list(name)  :=  [op(indets(sys,name))], n  ::integer  :=  1, {      rank  ::integer  :=  min (nops(sys),nops(vars)), maxminors  ::integer  :=  infinity }           )

Computes the nth iterated jacobian.

Parameters

Returns

A list of polynomials generating the nth element of the sequence (J_n) s.t.:

• J_0 = sys
• J_{n+1} = r x r minors of the Jacobian matrix of J_n w.r.t the variables vars

Example

> IteratedJacobian ([x^3,y^3], [x,y], 2);
                       3   3     2  2      4         4
                     [x , y , 9 x  y , 54 x  y, -54 y  x]

RealSolve := proc ( sys  ::{algebraic,list({algebraic,algebraic<algebraic})},    vars  ::list(name)  :=  [op(indets([sys,spec],name))], spec  ::seq(name=algebraic)  :=  NULL, {      noerror  ::truefalse  :=  false, precision  ::integer  :=  Digits }           )

Solve polynomial and trigonometric systems.

Parameters

Returns

A sequence of lists of the form v=n where v is a name and n is a numerical constant.