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Physicists computing multiloop Feynman diagrams have introduced various techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines

1) $DoT(r)=0$, r rational

2) $DoT(\pi^k)=k$, $k \in {\mathbb N}$,

3) $DoT(\zeta(k))=k$,

4) $DoT( a \cdot b)= DoT(a)+DoT(b)$

One then proves for example that the $\ell$-loop contribution to a certain scaling function in $N=4$ Supersymmetric gauge theory consists of a sum of terms all of which have DoT equal to $2 \ell-2$.

This can't be rigorous mathematically, since it is not even known that $\zeta(2n+1)$ is transcendental, but is there some circle of ideas, or conjecture in mathematics that if true would give a precise definition to DoT?

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    $\begingroup$ Not only that, but in what way is $\pi^k$ any more transcendental than $\pi?,$ given that it appears to be contained in $\mathbb{Z}[\pi]...$ $\endgroup$
    – Igor Rivin
    Commented Dec 27, 2010 at 3:57
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    $\begingroup$ Transcendentality ? I would have said transcendence. $\endgroup$ Commented Dec 27, 2010 at 4:01
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    $\begingroup$ You might be looking for some weight-like invariant of (periods of) motives, or of mixed Tate motives. For instance, the motive $\mathbb{Q}(-k)$ has period $\pi^k$ (up to algebraic multiples), and the weight of $\mathbb{Q}(-k)$ is $2k$. The paper "Motives associated to sums of graphs" by Spencer Bloch might point you in the right direction. (NB I've never read it.) For an introduction to periods, see the paper "Periods" by Kontsevich and Zagier. $\endgroup$
    – JBorger
    Commented Dec 27, 2010 at 4:56
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    $\begingroup$ The book "Feynman motives" by Matilde Marcolli might be useful as well. $\endgroup$ Commented Dec 27, 2010 at 6:54
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    $\begingroup$ Just to be clear, I did not invent the terminology and I have never been a disciple of the Maharishi Mahesh Yogi. $\endgroup$ Commented Dec 27, 2010 at 13:40

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Your transcendentality reminds me about the Institute of Algebraic Meditation at Höör (Sweden). To be honest, your definition corresponds to what is known as the weight of a (multiple) zeta value (see Michael Hoffman's http://www.usna.edu/Users/math/meh/mult.html, especially the references on MZVs). These indeed occur in the computation of Feynman's diagrams. As for conjectures related to the transcendental number theory tag, a belief is that $\pi$ and odd zeta values $\zeta(3)$, $\zeta(5)$, etc, are algebraically independent over the rationals.

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