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BÃ¼chiâ€™s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form xn + x 2 nâˆ’2 = 2x 2 nâˆ’1 + 2, n = 2, . . . , M âˆ’ 1, has no integer points other than those that satisfy Â±xn = Â±x0 +n (the Â± signs are independent). If answered positively, it would imply that there is no algorithm which decides, given anâ€¦ (More)

- Leonard Lipshitz, Thanases Pheidas
- J. Symb. Log.
- 1995

- Thanases Pheidas, Karim Zahidi
- J. Symb. Log.
- 2004

We generalize a question of BÃ¼chi : Let R be an integral domain, C a subring and k â‰¥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the kâˆ’th powers of the unknowns, with coefficients in C? We state a number-theoretical problem, depending on k, a positive answer toâ€¦ (More)

We prove that Hilbertâ€™s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called divisionample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.

- Thanases Pheidas
- J. Symb. Log.
- 1994

Lectures Two types of lectures were programmed into the schedule. First there were a number of introductory lectures which were were meant to introduce non-specialist into the area of Hilbertâ€™s Tenth Problem. Since quite a few of the participants came from different backgrounds this was an ideal way to get a feeling of the problems which come up in thisâ€¦ (More)

BÃ¼chiâ€™s problem is a number theoretical question, a positive answer to which would imply the following strengthening of the negative answer to Hilbertâ€™s Tenth Problem: the positive existential theory of the rational integers in the language of addition and a predicate for the property â€˜x is a squareâ€™ would be undecidable. BÃ¼chiâ€™s problem remains open. Inâ€¦ (More)

- Thanases Pheidas, Xavier Vidaux
- CiE
- 2005

Let F be a field of zero characteristic. We give the following answer to a generalization of a problem of BÃ¼chi over F [t]: A sequence of 92 or more cubes in F [t], not all constant, with third difference constant and equal to 6, is of the form (x+ n), n = 0, . . . , 91, for some x âˆˆ F [t] (cubes of successive elements). We use this, in conjunction to theâ€¦ (More)

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