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Keywords:
linear derivation; ring of constant; Fermat ring; Darboux polynomial; simple derivation
linear derivation; ring of constant; Fermat ring; Darboux polynomial; simple derivation
Summary:
At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer $w > 1 $ such that the polynomial ring in $n$ variables is $w$-differentially simple, all $w$ derivations are nonsimple and the $w$ derivations set contains a linear derivation.
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