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 . 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.