Real tropicalization and negative faces of the Newton polytope
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Real tropicalization and negative faces of the Newton polytope. / Telek, Máté L.
In: Journal of Pure and Applied Algebra, Vol. 228, No. 6, 107564, 2024.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Real tropicalization and negative faces of the Newton polytope
AU - Telek, Máté L.
N1 - Publisher Copyright: © 2023 The Author(s)
PY - 2024
Y1 - 2024
N2 - In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f1<0,…,fk<0} defined in the positive orthant and the combinatorial properties of the defining polynomials f1,…,fk. We describe a cone that depends only on the face structure of the Newton polytopes of f1,…,fk and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if S={f<0} and the polynomial f has generic coefficients. Furthermore, we show that for a maximally sparse polynomial f the real tropicalization of S={f<0} is determined by the outer normal cones of the Newton polytope of f and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant.
AB - In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f1<0,…,fk<0} defined in the positive orthant and the combinatorial properties of the defining polynomials f1,…,fk. We describe a cone that depends only on the face structure of the Newton polytopes of f1,…,fk and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if S={f<0} and the polynomial f has generic coefficients. Furthermore, we show that for a maximally sparse polynomial f the real tropicalization of S={f<0} is determined by the outer normal cones of the Newton polytope of f and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant.
KW - Logarithmic limit set
KW - Semi-algebraic set
KW - Signed support
KW - Signomial
U2 - 10.1016/j.jpaa.2023.107564
DO - 10.1016/j.jpaa.2023.107564
M3 - Journal article
AN - SCOPUS:85177849957
VL - 228
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
SN - 0022-4049
IS - 6
M1 - 107564
ER -
ID: 380352938