ABC: source and evidence reader

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \clearpage
2 \section{The universal mixed-weight jet filter}\label{sec:jet}
3 \subsection{One polynomial chooses the cancelling coefficients}
4 Let $m\ge0$ be an integer. Define
5 \[
6  P_m(X)=X\prod_{j=1}^m(jX-1)=\sum_{s=1}^{m+1}c_{m,s}X^s,
7 \]
8 and the formal series
9 \[
10  \Phi_m(t)=\frac1{(m+1)!}\sum_{s=1}^{m+1}c_{m,s}\Li_s(t).
11 \]
12 Here $\Li_s(t)=\sum_{k\ge1}t^k/k^s$ is first interpreted in $\Q[[t]]$. It also defines the usual analytic polylogarithm on $|t|<1$~\cite{DLMFPolylog}. The label $s$ is its weight. The contact order will follow from its coefficients, rather than from that label.
13 
14 \begin{theorem}[Exact jet cancellation]\label{thm:jet-contact}
15 \[
16  \Phi_m(t)=(-1)^m\sum_{k\ge1}\binom{k}{m+1}\frac{t^k}{k^{m+2}}.
17 \]
18 Thus
19 \[
20  \ord_0\Phi_m=m+1,\qquad
21  [t^{m+1}]\Phi_m=\frac{(-1)^m}{(m+1)^{m+2}}.
22 \]
23 \end{theorem}
24 \begin{proof}
25 The coefficient of $t^k$ is $P_m(1/k)/(m+1)!$. Direct substitution gives
26 \[
27  P_m(1/k)=\frac1k\prod_{j=1}^m\left(\frac jk-1\right)
28  =(-1)^m\frac{(k-1)\cdots(k-m)}{k^{m+1}}.
29 \]
30 Divide by $(m+1)!$ and multiply numerator and denominator by $k$ to get the asserted binomial coefficient. It vanishes for $1\le k\le m$ and first becomes one at $k=m+1$. The coefficient there is nonzero.
31 \end{proof}
32 For $m=0$, this is $\Phi_0=\Li_1$. For $m=1$,
33 \[
34  \Phi_1=(\Li_2-\Li_1)/2=-t^2/8-t^3/9-\cdots.
35 \]
36 The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coefficient.
37 
38 \subsection{Small coefficient norm has a limited meaning}
39 The coefficients of $P_m$ alternate in sign. Its absolute-coefficient polynomial evaluated at one gives
40 \[
41  \sum_s|c_{m,s}|=\prod_{j=1}^m(j+1)=(m+1)!.
42 \]
43 Also $c_{m,1}=(-1)^m$, so the integer coefficient vector is primitive. After the displayed normalization, its coefficient $\ell^1$ norm is one.
44 
45 These are exact statements about a vector of rational coefficients. The number of participating weights grows with $m$. A norm of one for the coefficient vector does not bound the denominator, covolume or arithmetic height of a period realization. No such realization is built into the definition of $\Phi_m$.
46 
47 \subsection{The depleted local series}
48 For a prime $p$, delete the terms whose index is divisible by $p$:
49 \[
50  \Phi_{m,p}(t)=(-1)^m\sum_{p\nmid k}\binom{k}{m+1}\frac{t^k}{k^{m+2}}\in\Z_p[[t]].
51 \]
52 It converges when $v_p(t)>0$.
53 \begin{theorem}[Exact local jet valuation]
54 If $v_p(t)=e\ge1$, then
55 \[
56  v_p(\Phi_{m,p}(t))=
57  \begin{cases}
58  (m+1)e,&p\nmid m+1,\\
59  (m+2)e,&p\mid m+1.
60  \end{cases}
61 \]
62 \end{theorem}
63 \begin{proof}
64 If $p\nmid m+1$, the first nonzero coefficient remains the unit $(-1)^m/(m+1)^{m+2}$. Its term has valuation $(m+1)e$. Every later coefficient is $p$-integral, so every later term has strictly larger valuation.
65 
66 If $p\mid m+1$, that first index is deleted. The next index is $m+2$, which is a $p$-unit. Its coefficient is $(-1)^m/(m+2)^{m+1}$, also a unit. All later terms again have strictly larger valuation. The ultrametric inequality with a unique least-valuation term proves both equalities.
67 \end{proof}
68 
69 \subsection{An exact differential identity}
70 Let $D=t\,\dd/\dd t$ be the Euler operator. Termwise formal differentiation is valid. Applying $D^{m+2}$ multiplies the coefficient at $t^k$ by $k^{m+2}$, so
71 \[
72  D^{m+2}\Phi_m=(-1)^m\sum_{k\ge1}\binom{k}{m+1}t^k
73  =(-1)^m\frac{t^{m+1}}{(1-t)^{m+2}}.
74 \]
75 The final equality is the elementary generating function for the binomial coefficients.
76 
77 This universal formal identity and the prime-dependent depleted valuations can be studied simultaneously. They do not identify the different $\Phi_{m,p}$ as local values of one algebraic number. A product formula cannot be applied merely because these functions arose from one coefficient pattern.

Back to source controls

Back to ABC: source and evidence reader