Some Remarks on Watanabe's Bold Conjecture

TL;DR

This paper investigates Watanabe's conjecture that all graded Artinian complete intersection algebras can embed into quadratic ones, confirming it for cases where defining polynomials split into linear factors.

Contribution

The paper proves Watanabe's conjecture for a specific class of complete intersection algebras with split linear factors.

Findings

Conjecture verified for algebras with splitting linear factors.

Supports the broader validity of Watanabe's conjecture.

Provides a partial proof in the context of quadratic embeddings.

Abstract

At the 2015 Workshop on Lefschetz Properties of Artinian Algebras, Junzo Watanabe conjectured that every graded Artinian complete intersection algebra with the standard grading can be embedded into another such algebra cut out by quadratic generators. We verify this conjecture in the case where the defining polynomials split into linear factors.