Again its (tautological) integer aﬃne-linear structure encodes the tropical structure on TPn .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
The complement TPn r Rn consists of (n + 1) divisors called the boundary divisors and isomorphic to TPn−1 .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
Points of sedentarity k are thus divided into (cid:18)n + 1 k (cid:19) strata that correspond to the k -dimensional faces of the (topological) n-simplex TPn .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
We may deﬁne the pro jective degree of a k -cycle A ⊂ TPn by computing its intersection number with one of the boundary TPn−k .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
Instead of giving the deﬁnition of the intersection number in this boundary case (which is also possible, but more complicated) we may choose yet another (non-boundary and easier to use) representative of TPn−k .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
We denote the union of all the closure of these quadrants in with Ln−k ⊂ TPn .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
Note that any translation in Rn may be extended to TPn by taking the closure.
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
Note that the subspace Lk is not even homeomorphic to TPn .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
In tropical geometry the space |D | is also kind of pro jective, though usually it is not isomorphic to TPn for any n.
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
To simplify our considerations we assume for the rest of these lectures that X = TPn .
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
We denote the corresponding set of tropical morphisms with Mg ,k ,d(TPn ).
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
Let us consider a the closure in Mg ,k ,d(TPn ) tropical morphisms that are realizable by topological immersions that locally deform in exactly ((n + 1)d + (n − 3)(1 − g ))-dimensional space.
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
Let us ﬁrst recall that, given a differentiable manifold N ⊆ Rn , a continuous map w : N → Rn with the property that for any p ∈ N , w(p) belongs to the tangent space TpN of N at p is called a tangent vector ﬁeld on N .
Harmonic solutions to a class of differential-algebraic equations with separated variables
Recall that if w : N → Rn is a tangent vector ﬁeld on the differentiable manifold N ⊆ Rn which is (Fr´echet) differentiable at p ∈ N and w(p) = 0, then the differential dpw : TpN → Rn maps TpN into itself (see, e.g. ), so that, the determinant det dpw is deﬁned.
Harmonic solutions to a class of differential-algebraic equations with separated variables
In the case when p is a nondegenerate zero (i.e. dpw : TpN → Rn is injective), p is an isolated zero and det dpw 6= 0.
Harmonic solutions to a class of differential-algebraic equations with separated variables
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