how do we prove that a sum of two periods is still a period?optimizing Frobenius instance solutionsDoes “all points rational” imply “constant” for this “cubic” curve over an arbitrary field?What is the relationship between these two notions of “period”?Computer software for periodsIs special value of Epstein zeta function in 3 variables a period?Property of a derivative in global fieldWhy are Green functions involved in intersection theory?Waldspurger Formula as a Torus IntegralHow does $zeta^mathfrakm(2)$ and relate to $zeta(2)$?Is there an algorithm for numerical approximation of (naive) period integrals
how do we prove that a sum of two periods is still a period?
optimizing Frobenius instance solutionsDoes “all points rational” imply “constant” for this “cubic” curve over an arbitrary field?What is the relationship between these two notions of “period”?Computer software for periodsIs special value of Epstein zeta function in 3 variables a period?Property of a derivative in global fieldWhy are Green functions involved in intersection theory?Waldspurger Formula as a Torus IntegralHow does $zeta^mathfrakm(2)$ and relate to $zeta(2)$?Is there an algorithm for numerical approximation of (naive) period integrals
$begingroup$
Kontsevich and Zagier define periods as the values of absolutely convergent integrals $int_sigma f$ where $f$ is a rational function with rational coefficients and $sigma$ is a semi-algebraic subset of $mathbbR^n$. How do we prove that the sum of two such numbers is still of this form? I've tried a few things but they don't seem to work...
ag.algebraic-geometry nt.number-theory
asked 2 days ago
periodsperiods
442
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Kontsevich and Zagier define periods as the values of absolutely convergent integrals $int_sigma f$ where $f$ is a rational function with rational coefficients and $sigma$ is a semi-algebraic subset of $mathbbR^n$. How do we prove that the sum of two such numbers is still of this form? I've tried a few things but they don't seem to work...
ag.algebraic-geometry nt.number-theory
asked 2 days ago
periodsperiods
442
New contributor
periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Kontsevich and Zagier define periods as the values of absolutely convergent integrals $int_sigma f$ where $f$ is a rational function with rational coefficients and $sigma$ is a semi-algebraic subset of $mathbbR^n$. How do we prove that the sum of two such numbers is still of this form? I've tried a few things but they don't seem to work...
ag.algebraic-geometry nt.number-theory
asked 2 days ago
periodsperiods
442
New contributor
periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Kontsevich and Zagier define periods as the values of absolutely convergent integrals $int_sigma f$ where $f$ is a rational function with rational coefficients and $sigma$ is a semi-algebraic subset of $mathbbR^n$. How do we prove that the sum of two such numbers is still of this form? I've tried a few things but they don't seem to work...
ag.algebraic-geometry nt.number-theory
ag.algebraic-geometry nt.number-theory
asked 2 days ago
periodsperiods
442
New contributor
periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
periodsperiods
442
New contributor
periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
periodsperiods
442
New contributor
periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
periodsperiods
442
asked 2 days ago
periodsperiods
442
442
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periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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periods is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1 Answer
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Let $alpha$ and $beta$ be two periods corresponding respectively to two absolutely convergent integrals $int_sigma f(x)dx$ and $int_tau g(y)dy$, where $f$ (resp. $g$) is a rational function on $Bbb Q$ with $r$ (resp. $s$) variables and $sigma$ (resp. $tau$) is a semi-algebraic subset of $Bbb R^r$ (resp. $Bbb R^s$).
Setting $omega:=sigmatimesleftlbrace0rightrbracetimes(0,1)^scoprod(0,1)^rtimesleftlbrace1rightrbracetimestau$, one immediately gets that $$alpha+beta=int_omega left[(1-t)f(x)+tg(y)right]dxdydt$$which is again an absolutely convergent integral, so that $alpha+beta$ is a period.
answered 2 days ago
GaussianGaussian
35318
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@periods: if you're not satisfied by the answer, please tell me how to improve it.
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– Gaussian
2 days ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Let $alpha$ and $beta$ be two periods corresponding respectively to two absolutely convergent integrals $int_sigma f(x)dx$ and $int_tau g(y)dy$, where $f$ (resp. $g$) is a rational function on $Bbb Q$ with $r$ (resp. $s$) variables and $sigma$ (resp. $tau$) is a semi-algebraic subset of $Bbb R^r$ (resp. $Bbb R^s$).
Setting $omega:=sigmatimesleftlbrace0rightrbracetimes(0,1)^scoprod(0,1)^rtimesleftlbrace1rightrbracetimestau$, one immediately gets that $$alpha+beta=int_omega left[(1-t)f(x)+tg(y)right]dxdydt$$which is again an absolutely convergent integral, so that $alpha+beta$ is a period.
answered 2 days ago
GaussianGaussian
35318
$endgroup$
$begingroup$
@periods: if you're not satisfied by the answer, please tell me how to improve it.
$endgroup$
– Gaussian
2 days ago
add a comment |
$begingroup$
Let $alpha$ and $beta$ be two periods corresponding respectively to two absolutely convergent integrals $int_sigma f(x)dx$ and $int_tau g(y)dy$, where $f$ (resp. $g$) is a rational function on $Bbb Q$ with $r$ (resp. $s$) variables and $sigma$ (resp. $tau$) is a semi-algebraic subset of $Bbb R^r$ (resp. $Bbb R^s$).
Setting $omega:=sigmatimesleftlbrace0rightrbracetimes(0,1)^scoprod(0,1)^rtimesleftlbrace1rightrbracetimestau$, one immediately gets that $$alpha+beta=int_omega left[(1-t)f(x)+tg(y)right]dxdydt$$which is again an absolutely convergent integral, so that $alpha+beta$ is a period.
answered 2 days ago
GaussianGaussian
35318
$endgroup$
$begingroup$
@periods: if you're not satisfied by the answer, please tell me how to improve it.
$endgroup$
– Gaussian
2 days ago
add a comment |
$begingroup$
Let $alpha$ and $beta$ be two periods corresponding respectively to two absolutely convergent integrals $int_sigma f(x)dx$ and $int_tau g(y)dy$, where $f$ (resp. $g$) is a rational function on $Bbb Q$ with $r$ (resp. $s$) variables and $sigma$ (resp. $tau$) is a semi-algebraic subset of $Bbb R^r$ (resp. $Bbb R^s$).
Setting $omega:=sigmatimesleftlbrace0rightrbracetimes(0,1)^scoprod(0,1)^rtimesleftlbrace1rightrbracetimestau$, one immediately gets that $$alpha+beta=int_omega left[(1-t)f(x)+tg(y)right]dxdydt$$which is again an absolutely convergent integral, so that $alpha+beta$ is a period.
answered 2 days ago
GaussianGaussian
35318
$endgroup$
Let $alpha$ and $beta$ be two periods corresponding respectively to two absolutely convergent integrals $int_sigma f(x)dx$ and $int_tau g(y)dy$, where $f$ (resp. $g$) is a rational function on $Bbb Q$ with $r$ (resp. $s$) variables and $sigma$ (resp. $tau$) is a semi-algebraic subset of $Bbb R^r$ (resp. $Bbb R^s$).
Setting $omega:=sigmatimesleftlbrace0rightrbracetimes(0,1)^scoprod(0,1)^rtimesleftlbrace1rightrbracetimestau$, one immediately gets that $$alpha+beta=int_omega left[(1-t)f(x)+tg(y)right]dxdydt$$which is again an absolutely convergent integral, so that $alpha+beta$ is a period.
answered 2 days ago
GaussianGaussian
35318
answered 2 days ago
GaussianGaussian
35318
answered 2 days ago
GaussianGaussian
35318
answered 2 days ago
GaussianGaussian
35318
35318
$begingroup$
@periods: if you're not satisfied by the answer, please tell me how to improve it.
$endgroup$
– Gaussian
2 days ago
add a comment |
$begingroup$
@periods: if you're not satisfied by the answer, please tell me how to improve it.
$endgroup$
– Gaussian
2 days ago
$begingroup$
@periods: if you're not satisfied by the answer, please tell me how to improve it.
$endgroup$
– Gaussian
2 days ago
$begingroup$
@periods: if you're not satisfied by the answer, please tell me how to improve it.
$endgroup$
– Gaussian
2 days ago
add a comment |
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