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.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left[x^2+y^2-1\,y-2x+7\right]$

$\mathrm{sys}≔\left[x^2+y^2-1\,y-2x+7\right]$

$\mathrm{rc1}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,'\mathrm{normalized}'='\mathrm{yes}'\right)$

$\mathrm{rc1}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{rc1}\left[1\right]\,R\right)$

$\left[2x-y-7\,5y^2+14y+45\right]$

$\mathrm{R2}≔\mathrm{PolynomialRing}\left(\left[x\,y\right]\,7\right)$

$\mathrm{R2}≔\mathrm{polynomial\_ring}$

$\mathrm{rc\_mod}≔\mathrm{Triangularize}\left(\mathrm{sys}\,\mathrm{R2}\,'\mathrm{normalized}'='\mathrm{yes}'\right)$

$\mathrm{rc\_mod}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{rc\_mod}\left[1\right]\,R\right)$

$\left[x+3y\,y^2+2\right]$

$\mathrm{rc2}≔\mathrm{Lift}\left(\mathrm{sys}\,R\,\mathrm{rc\_mod}\left[1\right]\,5\,7\right)$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{rc2}\,R\right)$

$\left[2x-y-7\,5y^2+14y+45\right]$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\right]\,962592769\right)$

$R≔\mathrm{polynomial\_ring}$

$f≔yx+1$

$f≔yx+1$

$\mathrm{Lift}\left(\left[f\right]\,R\,\mathrm{Chain}\left(\left[x+1\right]\,\mathrm{Empty}\left(R\right)\,R\right)\,y\,1\right)$

$\left[x+\frac{1}{y}\right]$