Polynomial Minimum Root Separation Note to a Paper of Sm Rump
Polynomial Minimum Root Separation Note to a Paper of Sm Rump
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To quickly assess the difficulty of the text, read a short excerpt:
) Proof: Take a small positive « srjuh that tbs preceeding Corollary applies to the circle Ci+, . If this circle is atti acting fojiP, thrn by Lemma 2 it contains at least one ezero of P' and the present Lemma is proved. Otherwise the preceeding Corollary implies the existence of a k s. D-1 such that C ^ is attracting and contains at least 3 disti^~roots of /', and therefore at least two eroots of P' (counted with mulHpUpi^. ) It follows that C y.^i contains at least this many eroots of P' . Q.... E. D.
Corollary 7: If p is a polynomial of degree d having distinct roots z^. Zj, then either the closed circle with i^. Zj contains an eroot of P', or the circle with center {z^+zjjfl and radius y~''-\z^-z2\ contains two eroots of P' (counted with multiplicity. ) Proof: Obvious from the preceeding Lemma.
The remainder of our argument follows that of [Ru79] quite closely. More specifically, we will prove: Theorem 8: Let /» be an arbitrary polynomial of degree d with integer coefficients. Then the minimum distance iep{P) between distinct roots z^, zi of P satisfies sep{P)^ (2^d(''*5K2(|/.
What to read after Polynomial Minimum Root Separation Note to a Paper of Sm Rump?
You can find similar books in the "Read Also" column, or choose other free books by Jacob T Schwartz to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
) Proof: Take a small positive « srjuh that tbs preceeding Corollary applies to the circle Ci+, . If this circle is atti acting fojiP, thrn by Lemma 2 it contains at least one ezero of P' and the present Lemma is proved. Otherwise the preceeding Corollary implies the existence of a k s. D-1 such that C ^ is attracting and contains at least 3 disti^~roots of /', and therefore at least two eroots of P' (counted with mulHpUpi^. ) It follows that C y.^i contains at least this many eroots of P' . Q.... E. D.
Corollary 7: If p is a polynomial of degree d having distinct roots z^. Zj, then either the closed circle with i^. Zj contains an eroot of P', or the circle with center {z^+zjjfl and radius y~''-\z^-z2\ contains two eroots of P' (counted with multiplicity. ) Proof: Obvious from the preceeding Lemma.
The remainder of our argument follows that of [Ru79] quite closely. More specifically, we will prove: Theorem 8: Let /» be an arbitrary polynomial of degree d with integer coefficients. Then the minimum distance iep{P) between distinct roots z^, zi of P satisfies sep{P)^ (2^d(''*5K2(|/.
What to read after Polynomial Minimum Root Separation Note to a Paper of Sm Rump?
You can find similar books in the "Read Also" column, or choose other free books by Jacob T Schwartz to read onlineMoreLess
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