Formal derivation

The formal derivation is a term from the mathematical branch of algebra . They transfer the derivation concept from analysis for functions to polynomials .

Since there is no number "between" two numbers over a ring , i.e. there is no concept of limit value , the difference quotient cannot be meaningfully defined and thus there is no derivation in the actual sense. In order to still be able to use the concept of derivation, it is formally defined for polynomials in such a way that the factor rule and the power rule are fulfilled.

definition

Let be a ring and denote the polynomial ring over in an indeterminate . For a polynomial ${\displaystyle R}$ ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle X}$

${\displaystyle f=\sum _{i=0}^{n}a_{i}X^{i}\in R[X]}$

the formal derivation is defined as ${\displaystyle f'\in R[X]}$

${\displaystyle f'=\sum _{i=1}^{n}ia_{i}X^{i-1}}$.

properties

• For the formal derivation, the well-known rules of differential calculus apply. In particular,

${\displaystyle (af+bg)'=af'+bg'}$ such as

${\displaystyle (fg)'=f'g+fg'}$

for everyone and everyone . That is, the figure ${\displaystyle f,g\in R[X]}$ ${\displaystyle a,b\in R}$

${\displaystyle D\colon R[X]\to R[X],\quad f\mapsto f'}$

is a derivation of . ${\displaystyle R[X]}$

• If in linear factors, ie , where the zeros are, then applies to the derivative ${\displaystyle f}$ ${\displaystyle f=\prod _{i=1}^{n}(X-r_{i})}$ ${\displaystyle r_{i}\in R}$ ${\displaystyle f}$

${\displaystyle f'=\sum _{i=1}^{n}\prod _{j=1 \atop j\neq i}^{n}(X-r_{j})}$.

application

If a body is a Euclidean ring (especially factorial ), where the Euclidean norm is used when the coefficients of are denoted by. The zeros of the GCF of and are precisely the multiple zeros of with an order lower by 1, as the following calculation shows: ${\displaystyle K}$ ${\displaystyle K[X]}$ ${\displaystyle \deg(f)=\max\{i\ |\ a_{i}\neq 0\}}$ ${\displaystyle a_{i}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f'}$ ${\displaystyle f}$

Let be a multiple zero of , then with a polynomial and a . It follows , so . ${\displaystyle r\in K}$ ${\displaystyle f\in K[X]}$ ${\displaystyle f=(X-r)((X-r)^{m}g)}$ ${\displaystyle g\in K[X]\setminus \{0\}}$ ${\displaystyle m\geq 1}$ ${\displaystyle f'=(X-r)^{m}g+(X-r)((X-r)^{m}g)'}$ ${\displaystyle f'(r)=0}$

literature

• Gerd Fischer : Textbook of Algebra. Vieweg, Wiesbaden 2008, ISBN 978-3-8348-0226-2 , pp. 275 ff. ( Limited preview in the Google book search).
• Christian Karpfinger, Kurt Meyberg: Algebra. Groups - rings - bodies. Spektrum Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2018-3 , p. 253 ff