Abstract

Answering a question by Nisse and Sottile, we derive a formula for the dimension of the amoeba of an irreducible algebraic variety.

Highlights

  • Introduction and main resultLet X ⊆ (C∗)n be an irreducible, closed algebraic subvariety

  • We prove an auxiliary result on subvarieties of real tori

  • The space on the left is contained in qpRn, and by Lemma 8, dqp abs maps it onto |qp|(ip−1TpZq) = q(ip−1TpZq)

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Summary

Introduction and main result

The Zariski-closure |X| in (R∗)n of the algebraic amoeba is stable under a subtorus of the real algebraic torus (R∗)n of dimension at least 2 dimC X − dimR A(X). If the amoeba has dimension strictly less than 2 dimC X, a positivedimensional real torus acts on |X| For any irreducible, closed subvariety X ⊆ (C∗) the dimension dimR A(X) is determined by the tropical variety Trop(X) ⊆ Rn of X via dimRA(X) = min{2 dimR Trop(X) + 2 dimR T − dimR S | T ⊆ S ⊆ Rn rational linear subspaces with S + (T + Trop(X)) = T + Trop(X)}, JAN DRAISMA, JOHANNES RAU AND CHI HO YUEN where a subspace of Rn is called rational if it is spanned by vectors in Qn. Similar to Theorem 3, we have the equivalent formula dimR A(X) = min{2 dimR(S + Trop(X)) − dimR S | S ⊆ Rn rational linear subspace}

In search of a positive-dimensional torus
Proofs of the main results
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