What does this research mean for the field?

Real tropicalization coincides with complex tropicalization for rank two and corank one symmetric matrices. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The research aims to explore the real and positive tropicalizations of low rank symmetric matrices over Puiseux series.

March 5, 2026Open Access

Real and Positive Tropicalizations of Symmetric Determinantal Varieties

Key Points

The research aims to explore the real and positive tropicalizations of low rank symmetric matrices over Puiseux series.
Analysis of real and complex tropicalization methods for rank two and corank one symmetric matrices.
Comparison of positive tropicalization definitions by Speyer and Williams.
Examination of symmetric rank two and corank one cases for discrepancies.
Real tropicalization aligns with complex tropicalization for rank two and corank one cases.
Positive tropicalization definitions coincide for symmetric rank two matrices.
Disparities exist in positive tropicalization for symmetric corank one matrices.

Abstract

We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.

Real and Positive Tropicalizations of Symmetric Determinantal Varieties

Key Points

Abstract

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