• DOI:10.37236/5232
• Corpus ID: 14173520

On Exponential Growth of Degrees

@article{Roichman2015OnEG,
  title={On Exponential Growth of Degrees},
  author={Yuval Roichman},
  journal={Electron. J. Comb.},
  year={2015},
  volume={22},
  pages={2}
}

• Yuval Roichman
• Published 3 May 2015
• Mathematics
• Electron. J. Comb.

A short proof to a recent theorem of Giambruno and Mishchenko is given in this note. 1 The theorem The following theorem was recently proved by Giambruno and Mishchenko. Theorem 1. [1, Theorem 1] For every 0 1 and n0 ∈ N, such that for every partition λ of n > n0 with max{λ1, λ1} β. The proof of Giambruno and Mishchenko is rather complicated and applies a clever order on the cells of the Young diagram. It should be noted that Theorem 1 is an immediate consequence of Rasala’s lower bounds on minimal degrees [2, Theorems F and H]. The proof of Rasala is very different and not less complicated; it relies heavily on his theory of degree polynomials. In this short note we suggest a short and relatively simple proof to Theorem 1. First, note that the following weak version is an immediate consequence of the hooklength formula. Lemma 2. The theorem holds for every 0 n! (2αn)n > ( e ) (2αn)n = β, where, by assumption, β := 1 2eα > 1. 2 Two lemmas Lemma 3. For every λ ` n ∏ (i,j)∈[λ] 1 h1,2 > · · · > h1,λ1 . Since h1,1 6 n it follows that ∏ (1,j)∈[λ] j6k h1j 6 (n)k