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When we compute a lex order Groebner basis, it is often efficient to
compute it via Groebner basis with respect to another order such as
degree reverse lex order, rather than to compute it directory by
`gr()`

etc. If we know that an input is a Groebner basis with
respect to an order, we can apply special methods called change of
ordering for a Groebner basis computation with respect to another
order, without using Buchberger algorithm. The following two functions
are ones for change of ordering such that they convert a Groebner
basis `gbase` with respect to the variable order `vlist1` and
the order type `order` into a lex Groebner basis with respect
to the variable order `vlist2`.

`tolex(`

`gbase`,`vlist1`,`order`,`vlist2`)-
This function can be used only when

`gbase`is an ideal over the rationals. The input`gbase`must be a Groebner basis with respect to the variable order`vlist1`and the order type`order`. Moreover the ideal generated by`gbase`must be zero-dimensional. This computes the lex Groebner basis of`gbase`by using the modular change of ordering algorithm. The algorithm first computes the lex Groebner basis over a finite field. Then each element in the lex Groebner basis over the rationals is computed with undetermined coefficient method and linear equation solving by Hensel lifting. `tolex_tl(`

`gbase`,`vlist1`,`order`,`vlist2`,`homo`)-
This function computes the lex Groebner basis of

`gbase`. The input`gbase`must be a Groebner basis with respect to the variable order`vlist1`and the order type`order`. Buchberger algorithm with trace lifting is used to compute the lex Groebner basis, however the Groebner basis check and the ideal membership check can be omitted by using several properties derived from the fact that the input is a Groebner basis. So it is more efficient than simple repetition of Buchberger algorithm. If the input is zero-dimensional, this function inserts automatically a computation of Groebner basis with respect to an elimination order, which makes the whole computation more efficient for many cases. If`homo`is not equal to 0, homogenization is used in each step.

For zero-dimensional systems, there are several functions to
compute the minimal polynomial of a polynomial and or a more compact
representation for zeros of the system. They are all defined in ‘`gr`’.
Refer to the sections for each functions.

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