Borel-fixed ideals and reduction number

The aim of this paper is to study the relationship between reduction numbers and Borel-fixed ideals in all characteristics. By definition, Borel-fixed ideals are closed under certain specializations which is similar to the strong stability. We will estimate the number of monomials which can be specialized to a given monomial. As a consequence, we obtain a combinatorial version of the well-known Eakin-Sathaye's theorem which bounds the reduction number in terms of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's theorem is attained by the reduction number of a lex-segment monomial ideal. This result answers a question of Conca in the affirmative. We will also show that the reduction number of the lex-segment ideal is bounded exponentially by the reduction number of the given ideal.Comment: 11 page

Do Convolutional Networks need to be Deep for Text Classification ?

We study in this work the importance of depth in convolutional models for text classification, either when character or word inputs are considered. We show on 5 standard text classification and sentiment analysis tasks that deep models indeed give better performances than shallow networks when the text input is represented as a sequence of characters. However, a simple shallow-and-wide network outperforms deep models such as DenseNet with word inputs. Our shallow word model further establishes new state-of-the-art performances on two datasets: Yelp Binary (95.9\%) and Yelp Full (64.9\%)