What bounds can we establish on coefficients of Swinnerton-Dyer polynomials?

The maximum coefficient of the nth Swinnerton-Dyer polynomial seems to grow very fast with n. These are the maximum absolute values of the first 6 polynomials,

1. 2
2. 10
3. 960
4. 13950764
5. 255690851718529024
6. 1771080720430629161685158978892152599456

What bounds can we establish on the absolute value of coefficients in the nth Swinnerton-Dyer polynomial? A very trivial bound appears to be $B(n) = 2^{2^n} n \sqrt{p_n}$ but this doesn't take into account cancellation of any of the terms. Is it possible to do better?

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Similar question recently posted to m.se: math.stackexchange.com/questions/192578/… – Gerry Myerson Sep 11 '12 at 22:36

There is a missing exponent in your trivial bound. Letting $u_n = \sum_{k=1}^n \sqrt{p_k}$, we have

$$B_0(n) \equiv \max_i \; \left| [x^i] S_n \right|$$ $$\le B_1(n) \equiv \max_i \; [x^i] (x+u_n)^{2^n} = \max_i {2^n \choose i} u_n^i$$ $$\le B_2(n) \equiv { 2^n \choose 2^{n-1} } u_n^{2^n}$$ $$\le B_3(n) \equiv 2^{2^n} (n \sqrt{p_n})^{2^n}.$$

Of course, in the last step we can get a better bound by estimating $u_n$ less crudely. It should also be possible to find an analytic bound for $B_1(n)$ that is less crude than $B_2(n)$ (regardless, $B_1(n)$ is easy to evaluate numerically).

For reference, I have computed numerical approximations of the actual value $B_0(n)$ up to $n = 20$ (correct up to rounding in the last digit):

0:  1
1:  2
2:  10
3:  960
4:  13950764
5:  2.55690851718529e+17
6:  1.77108072043063e+39
7:  8.57834471403602e+86
8:  4.69693103314689e+187
9:  3.24515842436673e+401
10: 8.31078370973184e+853
11: 4.18601441612784e+1793
12: 1.37441368638541e+3755
13: 7.32398012717744e+7815
14: 4.7364530607185e+16172
15: 8.41442697691365e+33355
16: 6.50154879984207e+68684
17: 2.98829955473397e+141188
18: 7.8161464597922e+289271
19: 2.1050522533847e+591950
20: 2.92232330678161e+1209132


Here is a comparison of the bounds:

It appears that $B_3$, $B_2$ and $B_1$ asymptotically overestimate the number of bits in $B_0$ less than by a respective factor 2.20, 2.02 and 1.76.

As you say, it should possible to do better by taking into account the cancellation that occurs. Perhaps by expressing the coefficients in terms of elementary symmetric polynomials? A lower bound would also be interesting.

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