lift a regular chain
Lift(F, R, rc, e, m)
list of polynomials of R
regular chain of R
positive integer or a variable of R
positive integer or a coefficient of R
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.
First we consider an example where coefficients are lifted.
R ≔ PolynomialRing⁡x,y
R ≔ polynomial_ring
sys ≔ x2+y2−1,y−2⁢x+7
rc1 ≔ Triangularize⁡sys,R,'normalized'='yes'
rc1 ≔ regular_chain
R2 ≔ PolynomialRing⁡x,y,7
R2 ≔ polynomial_ring
rc_mod ≔ Triangularize⁡sys,R2,'normalized'='yes'
rc_mod ≔ regular_chain
rc2 ≔ Lift⁡sys,R,rc_mod1,5,7
rc2 ≔ regular_chain
Next we consider an example where a variable is lifted.
R ≔ PolynomialRing⁡x,y,962592769
f ≔ y⁢x+1
f ≔ x⁢y+1
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