Definition of Tropical Hypersurface
$begingroup$
Given the tropical semiring $(mathbb{T},oplus,otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable.
I'm wondering how incidental the non-differentiable part is, and how this part is derived. For instance, had we given a different semiring $S$, would there be a canonical way of associating to $S$-polynomials $S$-hypersurfaces? I imagine this might have something to do with Universal Algebra, of which I am ignorant, so it would be good to get a light answer on this as much as possible, and if not possible a light reference would be greatly appreciated!
algebraic-geometry universal-algebra tropical-geometry
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
add a comment |
$begingroup$
Given the tropical semiring $(mathbb{T},oplus,otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable.
I'm wondering how incidental the non-differentiable part is, and how this part is derived. For instance, had we given a different semiring $S$, would there be a canonical way of associating to $S$-polynomials $S$-hypersurfaces? I imagine this might have something to do with Universal Algebra, of which I am ignorant, so it would be good to get a light answer on this as much as possible, and if not possible a light reference would be greatly appreciated!
algebraic-geometry universal-algebra tropical-geometry
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
add a comment |
$begingroup$
Given the tropical semiring $(mathbb{T},oplus,otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable.
I'm wondering how incidental the non-differentiable part is, and how this part is derived. For instance, had we given a different semiring $S$, would there be a canonical way of associating to $S$-polynomials $S$-hypersurfaces? I imagine this might have something to do with Universal Algebra, of which I am ignorant, so it would be good to get a light answer on this as much as possible, and if not possible a light reference would be greatly appreciated!
algebraic-geometry universal-algebra tropical-geometry
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
Given the tropical semiring $(mathbb{T},oplus,otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable.
I'm wondering how incidental the non-differentiable part is, and how this part is derived. For instance, had we given a different semiring $S$, would there be a canonical way of associating to $S$-polynomials $S$-hypersurfaces? I imagine this might have something to do with Universal Algebra, of which I am ignorant, so it would be good to get a light answer on this as much as possible, and if not possible a light reference would be greatly appreciated!
algebraic-geometry universal-algebra tropical-geometry
algebraic-geometry universal-algebra tropical-geometry
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363415
1,363415
add a comment |
add a comment |
1 Answer
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On not initially receiving an answer, I did some further reading on this and found a lovely light exposition: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf
The idea is to construct the tropical semiring $(mathbb{T},oplus,otimes)$ in analogy to the reals via closure of operations from the more primitive $(mathbb{N},oplus,otimes)$. What a root is turns out to be a pivotal concept. So the root of an $mathbb{R}$-monomial $P(x) = ax+b$ is a number $x_0$ for which $P(x_0)=0$, the additive identity. If we try transferring this over naively, we get the following:
$$ begin{align} a otimes x oplus b &= -infty \
iff max(a+x,b) &= - infty, end{align} $$
which is not analogously informative. In the above paper, they provide an analogue of the fundamental theorem of algebra, and give a justification for the 'non-differentiability' aspect as a consequence of wanting our $mathbb{T}$-roots to give us the information on the factorisation implied by the fundamental theorem.
So the associated hypersurface of a tropical polynomial is its set of tropical roots as defined above. Very happy to accept answers that might go into more detail or on anything I missed here!
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
add a comment |
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1 Answer
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$begingroup$
On not initially receiving an answer, I did some further reading on this and found a lovely light exposition: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf
The idea is to construct the tropical semiring $(mathbb{T},oplus,otimes)$ in analogy to the reals via closure of operations from the more primitive $(mathbb{N},oplus,otimes)$. What a root is turns out to be a pivotal concept. So the root of an $mathbb{R}$-monomial $P(x) = ax+b$ is a number $x_0$ for which $P(x_0)=0$, the additive identity. If we try transferring this over naively, we get the following:
$$ begin{align} a otimes x oplus b &= -infty \
iff max(a+x,b) &= - infty, end{align} $$
which is not analogously informative. In the above paper, they provide an analogue of the fundamental theorem of algebra, and give a justification for the 'non-differentiability' aspect as a consequence of wanting our $mathbb{T}$-roots to give us the information on the factorisation implied by the fundamental theorem.
So the associated hypersurface of a tropical polynomial is its set of tropical roots as defined above. Very happy to accept answers that might go into more detail or on anything I missed here!
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
add a comment |
$begingroup$
On not initially receiving an answer, I did some further reading on this and found a lovely light exposition: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf
The idea is to construct the tropical semiring $(mathbb{T},oplus,otimes)$ in analogy to the reals via closure of operations from the more primitive $(mathbb{N},oplus,otimes)$. What a root is turns out to be a pivotal concept. So the root of an $mathbb{R}$-monomial $P(x) = ax+b$ is a number $x_0$ for which $P(x_0)=0$, the additive identity. If we try transferring this over naively, we get the following:
$$ begin{align} a otimes x oplus b &= -infty \
iff max(a+x,b) &= - infty, end{align} $$
which is not analogously informative. In the above paper, they provide an analogue of the fundamental theorem of algebra, and give a justification for the 'non-differentiability' aspect as a consequence of wanting our $mathbb{T}$-roots to give us the information on the factorisation implied by the fundamental theorem.
So the associated hypersurface of a tropical polynomial is its set of tropical roots as defined above. Very happy to accept answers that might go into more detail or on anything I missed here!
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
add a comment |
$begingroup$
On not initially receiving an answer, I did some further reading on this and found a lovely light exposition: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf
The idea is to construct the tropical semiring $(mathbb{T},oplus,otimes)$ in analogy to the reals via closure of operations from the more primitive $(mathbb{N},oplus,otimes)$. What a root is turns out to be a pivotal concept. So the root of an $mathbb{R}$-monomial $P(x) = ax+b$ is a number $x_0$ for which $P(x_0)=0$, the additive identity. If we try transferring this over naively, we get the following:
$$ begin{align} a otimes x oplus b &= -infty \
iff max(a+x,b) &= - infty, end{align} $$
which is not analogously informative. In the above paper, they provide an analogue of the fundamental theorem of algebra, and give a justification for the 'non-differentiability' aspect as a consequence of wanting our $mathbb{T}$-roots to give us the information on the factorisation implied by the fundamental theorem.
So the associated hypersurface of a tropical polynomial is its set of tropical roots as defined above. Very happy to accept answers that might go into more detail or on anything I missed here!
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
$endgroup$
On not initially receiving an answer, I did some further reading on this and found a lovely light exposition: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf
The idea is to construct the tropical semiring $(mathbb{T},oplus,otimes)$ in analogy to the reals via closure of operations from the more primitive $(mathbb{N},oplus,otimes)$. What a root is turns out to be a pivotal concept. So the root of an $mathbb{R}$-monomial $P(x) = ax+b$ is a number $x_0$ for which $P(x_0)=0$, the additive identity. If we try transferring this over naively, we get the following:
$$ begin{align} a otimes x oplus b &= -infty \
iff max(a+x,b) &= - infty, end{align} $$
which is not analogously informative. In the above paper, they provide an analogue of the fundamental theorem of algebra, and give a justification for the 'non-differentiability' aspect as a consequence of wanting our $mathbb{T}$-roots to give us the information on the factorisation implied by the fundamental theorem.
So the associated hypersurface of a tropical polynomial is its set of tropical roots as defined above. Very happy to accept answers that might go into more detail or on anything I missed here!
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
edited Dec 4 '18 at 0:17
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363415
1,363415
add a comment |
add a comment |
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