Sendow's conjecture

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The Sendow conjecture , often referred to as the Ilieff conjecture or Ilieff-Sendow conjecture , is a previously unproven conjecture in function theory that was first established in 1958 by the Bulgarian mathematician Blagowest Sendow (1932-2020).

Content of Sendow's conjecture

It becomes a polynomial of the P th degree

considered with zeros from the unit circle of the complex number plane . These zeros do not have to be different.

Now the first derivative of the polynomial is formed. is a polynomial of degree . According to Gauss-Lucas's theorem , the zeros of this new polynomial all lie in the convex hull of the zeros of the polynomial , i.e. also within the unit circle.

The Sendowsche conjecture states that for every root of a zero point of are (that is, a critical point ), the distance is less than or equal to 1 to one another .

Although this assumption seems like a matter of course, it has not been fully proven until today (2020). Proofs were provided for polynomials up to degree 10. In addition, for polynomials of any degree it was proven that at least one zero of the first derivative of the polynomial is within a radius of 1.08331641 around a zero of the polynomial, i.e. with the above designations . A proof for polynomials of degree> 10 with radius = 1 around the zeros has not yet been found.

history

Blagowest Sendow expressed the presumption named after him in 1958 to Nikola Obreschkow , with whom he was a research assistant. In 1967, Walter Hayman erroneously referred to the Sendow conjecture as the Ilieff conjecture in his publication, and this variety of names has existed since then.

Many mathematicians around the world have tried to prove this conjecture, so far only with partial success.

In 1969 A. Meir and A. Sharma proved the conjecture for polynomials of degree less than 6. In 1985, B. Bojanov, Q. Rahman, and J. Szynal proved the conjecture for polynomials of any degree for a radius of 1.08331641. In 1991, J. Brown proved the conjecture for polynomials of degree less than 7. In 1996, I. Borcea proved the conjecture for polynomials with degree less than 8. In 1999, J. Brown and G. Xiang proved the conjecture for polynomials with degree less than 9. Zaizhao Meng succeeded in proving the conjecture for 9th degree polynomials in 2018. Dinesh Sharma Bhattarai provided the proof for 10th degree polynomials in 2019.

Individual evidence

  1. ↑ a b c d e f g The Ilieff-Sendov Conjecture at faculty.etsu.edu. Retrieved February 28, 2020.
  2. ↑ a b c d Sendov's conjecture - a current mathematical problem at presse.uni-oldenburg.de. Retrieved February 28, 2020.
  3. ↑ A Conjecture in the Geometry of Polynomials at kurims.kyoto-u.ac.jp. Retrieved February 28, 2020.
  4. ↑ a b Zaizhao Meng: Proof of the Sendov conjecture for polynomials of degree nine , 2018 online
  5. ↑ a b Dinesh Sharma Bhattarai: A Proof of Sendov's conjecture for Polynomials of degree Ten , International Journal of Scientific and Research Publications, Volume 9, Issue 8, August 2019, ISSN 2250-3153 online
  6. ↑ Благовест Сендов at mmib.math.bas.bg (Bulgarian). Retrieved February 28, 2020.
  7. ↑ Почина Академик Благовест Сендов - Световно Известен Математик at bulgarica.com. Retrieved February 28, 2020.
  8. ^ Walter Hayman: Research Problems in Function Theory , London: Athlone, 1967
  9. ↑ Meir, A. and A. Sharma: On Ilyeff's Conjecture,, Pacific Journal of Mathematics, 31, 459-467, 1969.
  10. ↑ Bojanov, B., Rahman, Q., and J. Szynal: On a Conjecture of Sendov about the Critical Points of a Polynomial , Mathematische Zeitschrift, 190, 281-285, 1985.
  11. ↑ Brown, J .: On the Sendov Conjecture for Sixth Degree Polynomials , Proceedings of the American mathematical Society, 113 (4), 939-946, 1991.
  12. ↑ Borcea, I .: On the Sendov Conjecture for Polynomials with at Most Six Distinct Roots , Journal of Mathematical Analysis and Applications, 200, 182-206, 1996.
  13. ↑ Brown, J. and G. Xiang: Proof of the Sendov Conjecture for Polynomials of Degree at Most Eight , Journal of Mathematical Analysis and Applications, 232, 272-292, 1999.

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