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Van Der Poorten conjectures that if a power serie over the rationals is the diagonal of a serie in two variables then it is algebraic over Q(X) iff it is algebraic over almost every reduction module p, and the degree is bounded independiently of p. I want to know if there are some results on these way and some references.

flag	asked Aug 30 at 16:22 camilo 19●1

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Could you give some more background on this? – Piotr Achinger Aug 30 at 19:02

In case when the series is a solution to some linear differential equation, this seems to be related to en.wikipedia.org/wiki/… . – Piotr Achinger Aug 30 at 19:04

Do you want you power series to have integer coefficients so you can reduce it mod primes? – Keenan Kidwell Aug 30 at 23:31

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Every power series $\sum a_n x^n$ is the diagonal of a series in two variables, e.g., $\sum a_n x^ny^n$. It is easy to give a nonalgebraic power series over the integers that is a polynomial (and hence algebraic of degree one) modulo any prime, e.g., $2 + 2\cdot 3x + 2\cdot 3\cdot 5 x^2+\cdots$. Hence I don't think that you have correctly stated Van Der Poorten's conjecture. – Richard Stanley Aug 31 at 0:33

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Based on your answer below, I assume you read French. If so, you might be better off asking questions in French (I think this is acceptable). Because either your English is woefully deficient, or you are seriously not understanding what you are talking about, judging from what I have seen so far from your MO contributions. I am sorry to have to say this so bluntly and harshly. – Todd Trimble Sep 2 at 4:56

camilo · Answer 1 · 2012-09-01 20:48:37Z UTC

the paper when it this cojecture is in the seminaire de theorie des nombres de paris 1990-1991 the name is power series representing algebraic functions, there are other paper of JP Allouche, the name is tracendence of formal series with rational coeficients. I belive that the degree that is bounded is about the degree of coefficients, i don´t understan well this fact, if you have more information about it I would thank to much.

Algebraicity of power series over the rationals from the algebraicity over Fp

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