56 research outputs found
On the grasshopper problem with signed jumps
The 6th problem of the 50th International Mathematical Olympiad (IMO), held
in Germany, 2009, was the following.
Let be distinct positive integers and let be a set of
positive integers not containing . A grasshopper is to
jump along the real axis, starting at the point 0 and making jumps to the
right with lengths in some order. Prove that the order can be
chosen in such a way that the grasshopper never lands on any point in .
The problem was discussed in many on-line forums, as well by communities of
students as by senior mathematicians. Though there have been attempts to solve
the problem using Noga Alon's famous Combinatorial Nullstellensatz, up to now
all known solutions to the IMO problem are elementary and inductive. In this
paper we show that if the condition that the numbers are positive
is omitted, it allows us to apply the polynomial method to solve the modified
problem.Comment: Submitted to AMS Monthly on 1st March, 2010; last revision in August,
201
Alon's Nullstellensatz for multisets
Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one
of the most powerful algebraic tools in combinatorics, with a diverse array of
applications. Let \F be a field, be finite nonempty
subsets of \F. Alon's theorem is a specialized, precise version of the
Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing
on the set S=S_1\times S_2\times ... \times S_n\subseteq \F^n. From this Alon
deduces a simple and amazingly widely applicable nonvanishing criterion
(Theorem 1.2 in \cite{Alon1}). It provides a sufficient condition for a
polynomial which guarantees that is not identically zero
on the set . In this paper we extend these two results from sets of points
to multisets. We give two different proofs of the generalized nonvanishing
theorem. We extend some of the known applications of the original nonvanishing
theorem to a setting allowing multiplicities, including the theorem of Alon and
F\"uredi on the hyperplane coverings of discrete cubes.Comment: Submitted to the journal Combinatorica of the J\'anos Bolyai
Mathematical Society on August 5, 201
Remarks to Arsovski's proof of Snevily's conjecture
Based on the recent work of Arsovski, we confirm a conjecture of Feng, Sun,
and Xiang, and we give a shortened proof of Snevily's conjecture.Comment: 5 pages, LaTeX2e; v2: revised version incorporating suggestions by
the referee (e.g. title was changed); to appear in Ann. Univ. Sci. Budapest.
E\"otv\"os Sect. Math
Some extensions of Alon's Nullstellensatz
Alon's combinatorial Nullstellensatz, and in particular the resulting
nonvanishing criterion is one of the most powerful algebraic tools in
combinatorics, with many important applications. In this paper we extend the
nonvanishing theorem in two directions. We prove a version allowing multiple
points. Also, we establish a variant which is valid over arbitrary commutative
rings, not merely over subrings of fields. As an application, we prove
extensions of the theorem of Alon and F\"uredi on hyperplane coverings of
discrete cubes.Comment: Inital submission: Thu, 24 Mar 201
Analízis feladatgyűjtemény II.
Ez a feladatgyűjtemény elsősorban azon egyetemi hallgatók számára készült, akik matematikát, ezen belül kalkulust és analízist tanulnak. A könyv fő feladata bevezetni az olvasót a a differenciál és integrálszámításba és ezek alkalmazásaiba
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