Bounds in tropical geometry: between complex and real

We show that bounds in tropical geometry can have for some problems features of
complex geometry and for other problems of real geometry.

We prove (jointly with V. Podolskii) a tropical dual effective
Nullstellensatz which states that a system of tropical polynomials
$f_1,\dots, f_k$ in $n$ variables of degrees $d_1\ge \cdots \ge d_k$
has a solution iff a finite (truncated) submatrix of (infinite)
Macauley matrix of the system has a tropical solution. The size of
truncation depends on the sum of degrees $d_1+\cdots + d_k$ in case
of a tropical semi-ring without infinity. In case of a tropical
semi-ring with infinity the size of truncation depends on the
product of degrees $d_1 \cdots d_n$ (Bezout number). The bounds are
close to sharp. This phenomenon resembles the classical effective
Nullstellensatz in the complex geometry for which a similar bound
being the sum of degrees holds in the projective case
(Macauley-Lazard), while the bound being the product of degrees
emerges in the affine case
(Brownawell-Galligo-Giusti-Heintz-Kollar). We consider the dual
Nullstellensatz since the classical Hilbert's one fails in the
tropical world.

On the other hand, the sum of Betti numbers of a tropical prevariety
given by $f_1,\dots, f_k$ is less than by ${k+n \choose n} \cdot
d_1\cdots d_n$ (a joint result with A. Davydow), which resembles
Oleinik-Petrovsky-Milnor-Thom bound  on the sum of Betti numbers of
a real semi-algebraic set. Again, this bound is close to sharp.

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