SENDOV'S CONJECTURE: RELATION BETWEEN ROOTS AND CRITICAL POINTS OF A COMPLEX POLYNOMIAL
Sendov’s Conjecture states that if all the roots of a complex polynomial P(z) lies within the unit disk, then each closed disk with unit radius and center at a root P(z) contains a critical point of P(z). In this final project, we shall consider some results on Sendov’s Conjecture which will show...
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| Format: | Final Project |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/77564 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | Sendov’s Conjecture states that if all the roots of a complex polynomial P(z) lies
within the unit disk, then each closed disk with unit radius and center at a root
P(z) contains a critical point of P(z). In this final project, we shall consider some
results on Sendov’s Conjecture which will show the relationship between the roots
and critical points of a complex polynomial. In the first chapter, we will give a
brief restatement of known results regarding roots and critical points for real and
complex polynomials. In the following chapter, we will give an introduction of
Sendov’s Conjecture along with some partial results on special cases. Then, using
partial results on the unit circle and origin, also using the rotation invariant propery,
we will give a restatement of Sendov’s Conjecture. Finally, in the third chapter, we
will be showing a result on Sendov’s Conjecture for high degree polynomials by
observing the properties of a complex polynomial assumed to contradict Sendov’s
Conjecture at a root a ? (0, 1). |
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