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)
Summary
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}
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