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
1,363515
$endgroup$
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
1,363515
$endgroup$
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
1,363515
$endgroup$
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
1,363515
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363515
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363515
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363515
asked Dec 2 '18 at 16:12
Andrew WhelanAndrew Whelan
1,363515
1,363515
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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
1,363515
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022824%2fdefinition-of-tropical-hypersurface%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
1,363515
$endgroup$
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
1,363515
$endgroup$
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
1,363515
$endgroup$
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
1,363515
edited Dec 4 '18 at 0:17
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363515
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363515
answered Dec 4 '18 at 0:03
Andrew WhelanAndrew Whelan
1,363515
1,363515
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022824%2fdefinition-of-tropical-hypersurface%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
