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m →Sturm chains: comma |
Corrected a mistake on the sign of p_1 and expanded a bit on what happens if the polynomial isn't square free |
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of decreasing [[degree of a polynomial|degree]] with these following properties:
* <math>p_0= p</math> is [[square-free polynomial|square free]];
* if <math>p(\xi)=0</math>, then <math>\operatorname{sign}(p_1(\xi))=
* if <math>p_i(\xi)=0</math>
* <math>p_m</math> does not change its sign.
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\end{align} </math>
That is, successively take the remainders with [[polynomial#Divisibility | polynomial division]] and change their signs. Since <math>\operatorname{deg}(p_{i+1}) < \operatorname{deg} (p_i)</math> for <math>0 \le i < m</math>, the algorithm terminates. The final polynomial, ''p''<sub>''m''</sub>, is the [[greatest common divisor]] of ''p'' and its derivative. Since ''p'' is square free, it has only has simple roots and shares no roots with its derivative, so ''p''<sub>''m''</sub> will be a non-zero constant. The Sturm chain then is
:<math>p,p_1,p_2,\ldots,p_m . \,</math>
Incidentally, if ''p''<sub>''m''</sub> turned out to be zero, then ''p'' would not be square free, but ''p''<sub>''m-1''</sub> would then be a non-trivial factor of ''p'', so we could divide it out and calculate Sturm chains for both ''p''<sub>''m-1''</sub> and ''p''/''p''<sub>''m-1''</sub> (the later of which is guaranteed to be square free). Repeating this process, we can obtain a square free factorization for ''any'' polynomial ''p'' and a Sturm chain associated with each square free factor.
==Statement==
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