# On a conjecture of V. V. Shchigolev

## TL;DR

This paper confirms Shchigolev's conjecture that any T-space generated by an infinite subset of a specific set G over fields of characteristic p>2 is finitely based, providing explicit bounds on the basis size.

## Contribution

The paper proves Shchigolev's conjecture, demonstrating that all T-spaces generated by infinite subsets of G are finitely based with bounded basis size.

## Key findings

- Shchigolev's conjecture is true for fields of characteristic p>2.
- T-spaces generated by infinite subsets of G have bases of size at most i_2 - i_1 + 1.
- Provides explicit bounds on the basis size for these T-spaces.

## Abstract

V. V. Shchigolev has proven that over any infinite field k of characteristic p>2, the T-space generated by G={x_1^p,x_1^px_2^p,...} is finitely based, which answered a question raised by A. V. Grishin. Shchigolev went on to conjecture that every infinite subset of G generated a finitely based T-space. In this paper, we prove that Shchigolev's conjecture was correct by showing that for any field of characteristic p>2, the T-space generated by any subset {x_1^px_2^p...x_{i_1}^p, x_1^px_2^p...x_{i_2}^p,...}, i_1<i_2<i_3<..., of G has a T-space basis of size at most i_2-i_1+1.