Dear All,

I have written a paper on finding (deterministically) an upper bound to the largest non-periodic zero of an integer linear recurrence, in some special cases.

Please go through my paper at https://vixra.org/abs/2308.0001, and feel free to offer your comments here.

Thanks,
-Deepak

Dear All,

Four updates on my paper, all of which will be addressed in the subsequent version:

UPDATE 1:
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The algorithms in the proofs in Theorems 6 and 7, currently have a minor repairable flaw. The repair is that the bound should also be calculated for the first theta if is a rational multiple of PI, and finally output the minimum(this first bound, the bound calculated in step 10). This will take care of the case where the first theta is a rational multiple of PI. Its just 2 additional steps:

Step 0: Set w_rational = value of n beyond which
(r1^n min_non_zero_absolute(cos(theta1 n)) absolute(d_0_1 + d_1_1 n + d_2_1 n^2 + ... + d_L-1_1 n^(L-1))) > SUM((r_k'^n (absolute(d_0_k') + absolute(d_1_k')n + absolute(d_2_k')n^2 + ... + absolute(d_L-1_k')n^(L-1))) , over integers k' in [2, L]).

Step 13: OUTPUT minimum(w_rational, w found in step 10)

Similar change to be made in algorithms of both Theorems 6 & 7.

UPDATE 2:
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It is easy to come out with an explicit closed-form-expression of the upper bounds mentioned in the algorithms 6 & 7. Only thing is that the closed-form expression of the bounds would be much larger than that obtained by the deterministic algorithm.

UPDATE 3:
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The name of the author in the first reference "H Derksen" has been wrongly autocorrected before my submission, which I did not notice. This will be corrected in the subsequent version of this paper.

UPDATE 4:
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There is other previously existing research (eg. 2015 paper of Min Sha "https://www.sciencedirect.com/science/article/abs/pii/S0022314X18302476") that introduces an algorithm for finding the upper bound for the largest non-periodic zero of a simple linear recurrence. "Simple" here means that the characteristic roots are unique. So I believe that one area where my paper is different (i.e. my contribution) is that it covers the case of repeated roots, and roots with equal moduli but arguments that are commensurable.

There is also other previously existing research covering cases of the degree of the linear recurrence being less than 5.

If you know of any other reseach papers on this topic, please let me know and I will be happy to read them.

> Step 13: OUTPUT minimum(w_rational, w found in step 10)

Its "maximum" not minimum

Dear All,

The updated version of my paper with enhanced explanations on all the theorems, is now live at https://vixra.org/abs/2308.0001

I look forward to your comments.

Thank you,
-Deepak