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RegularChains[ChainTools][Lift] - lift a regular chain

Calling Sequence

Lift(F, R, rc, e, m)

Parameters

F

-

list of polynomials of R

R

-

polynomial ring

rc

-

regular chain of R

e

-

positive integer or a variable of R

m

-

positive integer or a coefficient of R

Description

• 

The function Lift returns a lifted regular chain over R lifted from the regular chain rc.  That is, the output regular chain reduces every polynomial of F to zero over R as long as rc does so under an image of R. This command works for two types of images of R.  If R has characteristic 0, it lifts from an image of finite characteristic.  If R has finite characteristic, it lifts from an image with fewer variables.

• 

More precisely, if e is a positive integer, then the base field is assumed to be the field of rational numbers and rc reduces every polynomial of F modulo m, where m is greater than or equal to 2. If e is a variable, then the base field is assumed to be a prime field and rc reduces every polynomial of F modulo e-m, where e is a variable of R and m belongs to the base field.

• 

In both cases, F must be a square system, that is the number of variables of R is equal to the number of elements of F. In the former case, rc and e-m must form a zero-dimensional normalized regular chain which generates a radical ideal in R. In the latter case, rc must be a zero-dimensional normalized regular chain which generates a radical ideal in R modulo m.

• 

In the former case, the Jacobian matrix of F must be invertible modulo rc and e-m. In the latter case, the Jacobian matrix of F must be invertible modulo rc and m.

• 

The function uses Hensel lifting techniques. If e is a positive integer then e is used as an upper bound on the number of lifting steps.

• 

For the case where e is variable, FFT polynomial arithmetic is used. This implies that the ring R should satisfy the hypotheses of the commands from the FastArithmeticTools subpackage.

Examples

withRegularChains:

withChainTools:

First we consider an example where coefficients are lifted.

R:=PolynomialRingx,y

R:=polynomial_ring

(1)

sys:=x2+y21,y2x+7

sys:=x2+y21,y2x+7

(2)

rc1:=Triangularizesys,R,'normalized'='yes'

rc1:=regular_chain

(3)

Equationsrc11,R

2xy7,5y2+14y+45

(4)

R2:=PolynomialRingx,y,7

R2:=polynomial_ring

(5)

rc_mod:=Triangularizesys,R2,'normalized'='yes'

rc_mod:=regular_chain

(6)

Equationsrc_mod1,R

x+3y,y2+2

(7)

rc2:=Liftsys,R,rc_mod1,5,7

rc2:=regular_chain

(8)

Equationsrc2,R

2xy7,5y2+14y+45

(9)

Next we consider an example where a variable is lifted.

R:=PolynomialRingx,y,962592769

R:=polynomial_ring

(10)

f:=yx+1

f:=xy+1

(11)

Liftf,R,Chainx+1,EmptyR,R,y,1

x+1y

(12)

See Also

Equations, FastArithmeticTools, JacobianMatrix, MatrixInverse, PolynomialRing, RegularChains, Triangularize


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