compute the common solutions of a polynomial and a regular chain - Maple Help

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RegularChains[Intersect] - compute the common solutions of a polynomial and a regular chain

Calling Sequence

Intersect(f, rc, R)

Parameters

f

-

polynomial

rc

-

regular chain

R

-

polynomial ring

Description

• 

The command Intersect(f, rc, R) computes the common solutions of the polynomial f and the regular chain rc in the following sense. Let V be the hypersurface defined by f, that is, the solutions of the equation f = 0 . Let W be the quasi-component of rc. Then Intersect(f, rc, R) returns regular chains such that the union of their quasi-components contains the intersection of V and W, and this union is contained in the intersection of V and the Zariski closure of W. See ConstructibleSetTools for a definition of a quasi-component.

• 

When the regular chain rc has dimension zero, Intersect(f, rc, R) computes exactly the intersection of V and W. This is also the case when W is a variety (that is a closed set for Zariski topology) or when rc has dimension one and f is regular w.r.t. the saturated ideal of rc. In all other cases, Intersect(f, rc, R) computes a superset of the intersection of V and W. However this superset is very close to this intersection.

• 

In summary and in broad terms, Intersect(f, rc, R) computes a sharp approximation of the intersection of V and W by means of regular chains.

• 

You can use the function Intersect to solve systems of equations incrementally, that is, one equation after the other. The example below illustrates this strategy.

• 

Another way of understanding the Intersect command is to observe that it specializes the solutions of rc with the constraint f = 0 .

Examples

withRegularChains:

withChainTools:

Define a ring of polynomials.

vars:=x,y,z:R:=PolynomialRingvars:

Define a set of equations.

sys:=x2+y+z1,x+y2+z1,x+y+z21

sys:=x2+y+z1,y2+x+z1,z2+x+y1

(1)

Define the empty regular chain.

rc:=EmptyR

rc:=regular_chain

(2)

Solve the first equation.

dec:=Intersectsys1,rc,R;mapEquations,dec,R

dec:=regular_chain

x2+y+z1

(3)

Solve the first and second equations.

dec:=seqopIntersectsys2,rc,R,rc=dec;mapEquations,dec,R

dec:=regular_chain,regular_chain

xy,y2+y+z1,x+y1,y2y+z

(4)

Solve the three equations together.

dec:=seqopIntersectsys3,rc,R,rc=dec;mapEquations,dec,R

dec:=regular_chain,regular_chain,regular_chain,regular_chain

xz,yz,z2+2z1,x,y,z1,x1,y,z,x,y1,z

(5)

See Also

ConstructibleSetTools, GeneralConstruct, RegularChains, Triangularize

References

  

Moreno Maza, M. "On Triangular Decompositions of Algebraic Varieties." MEGA-2000 conference. Bath, UK, England.


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