3 mathematical new results from 2019-2020 years.

in Education6 months ago (edited)
  1. Counterexample to Hedetniemi's conjecture.

https://arxiv.org/abs/1905.02167

The problem in math is that oftenly easy statements have hard proves whereas hard statements have easy proves. Hedetniemi's conejcture is of first kind. Hedetniemi conjecture stated that chromatic number of product of graphs GxH is equal to the smaller of chromatic numbers of G and H.
Yaroslav Shitov proved it is not true. The paper is quite hard, even for PhD I had graph seminar with last year.

  1. Mn(F2) is nil-clean for n=8k+4.

https://www.researchgate.net/publication/332630367_The_ring_mathrmM_8k4mathbbZ_2_is_nil-clean_of_index_four

One page publication - that cannot happen in biophysics, bioinformatics, chemistry, physics, astrology, biology, geography, history - it can happen only in math :).
Exercise: Why that method does not work for even number of copies of that matrix?
If matrix 8x8 has decomposition (P,Q), shouldn't its restriction to 4x4 minor in left upper corner be the decomposition of A? When we multiply block matrix, we multiply it's blocks, as they don't interfere?
I give 10 HIVE to the person who writes in comment proper reason why it can't be generalised such way (I understand why).
Even if You can't, read once again who wrote that paper and look back at 1.

  1. New faster decomposition into prime factors.

https://arxiv.org/abs/2010.05450

That paper was published 12 October, 2020!
Coronamadness does not stop science. (By the way, I can write in the future post about results of Soviet Mathematicians during 1941-1945, they were writing math papers even in besieged Leningrad!).

The naive deterministic algorithm of decomposition is of order N^(1/2).
In 1970's, it was N^(1/3).
A few years ago it was N^(1/4 + o(1)).
Hittmeir recently upgraded it to N^(2/9 +o(1))
Not Harvey upgraded Hittmeir's version to N^(1/5 + o(1)) and suggests that N^(1/6 + o(1)) is possible.
Do You know why in number theory algorithms working in time like N^1/5 are exponnetial time algorithms, not polynomial time?
3 HIVE for first person who explains in the comment.

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