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Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers
"... Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of . ..."
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Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of .
Ranks of twists of elliptic curves and Hilbert’s tenth problem, arxiv:0904.3709v2 [math.NT
"... Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bound ..."
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Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial MordellWeil group, and (assuming the ShafarevichTate conjecture) many others with infinite cyclic MordellWeil group. Using work of Poonen and Shlapentokh, it follows from our results that if the ShafarevichTate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field. 1. Introduction and
Hilbert’s Tenth Problem for algebraic function fields of characteristic 2
 Pacific J. Math
, 2003
"... Let K be an algebraic function field of characteristic 2 with constant field CK. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over CK and x algebraic over C(u) and such that K = CK(u, ..."
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Let K be an algebraic function field of characteristic 2 with constant field CK. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over CK and x algebraic over C(u) and such that K = CK(u, x). Then Hilbert’s Tenth Problem over K is undecidable. Together with Shlapentokh’s result for odd characteristic this implies that Hilbert’s Tenth Problem for any such field K of finite characteristic is undecidable. In particular, Hilbert’s Tenth Problem for any algebraic function field with finite constant field is undecidable. 1. Introduction. Hilbert’s Tenth Problem in its original form can be stated in the following form: Is there a uniform algorithm that determines, given a polynomial equation with integer coefficients, whether the equation has an integer solution
Hilbert's Tenth Problem over Rings of NumberTheoretic Interest
, 2003
"... This article is a survey about analogues of Hilbert's Tenth Problem over various rings, especially rings of interest to number theorists and algebraic geometers. For more details about most of the topics considered here, the conference proceedings [DLPVG00] is recommended. 2. The original problem H ..."
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This article is a survey about analogues of Hilbert's Tenth Problem over various rings, especially rings of interest to number theorists and algebraic geometers. For more details about most of the topics considered here, the conference proceedings [DLPVG00] is recommended. 2. The original problem Hilbert's Tenth Problem (from his list of 23 problems published in 1900) asked for an algorithm to decide whether a diophantine equation has a solution. More precisely, the input and output of such an algorithm were to be as follows: input: a polynomial f(x 1 , . . . , x n ) having coe#cients in Z Date: February 28, 2003
Part I:
, 2010
"... 1 Write down a random polynomial equation in two or more variables with coefficients in the ring of integers, e.g., 3X 3 + 4Y 3 + 5Z 3 = 0 and—chances are — it will be very tricky to find all its solutions; often you will be quite challenged by the question of whether or not it has solutions. Hilber ..."
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1 Write down a random polynomial equation in two or more variables with coefficients in the ring of integers, e.g., 3X 3 + 4Y 3 + 5Z 3 = 0 and—chances are — it will be very tricky to find all its solutions; often you will be quite challenged by the question of whether or not it has solutions. Hilbert spurred mathematicians to systematically investigate the general question: How solvable are such Diophantine equations? I will talk about this, and its relevance to specific number theoretic projects, and then aim towards some recent work, joint with Karl Rubin. Here is a close translation of Hilbert’s formulation of the problem: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. I wonder what Hilbert meant by a Diophantine equation and process. There is a vagueness in the quantification: are we allowed a different algorithm for each equation, are we expected to find single processes
PROBLEMS CONNECTING LOGIC AND NUMBER THEORY
"... asked to take part in a questionandanswer session with Carol Wood and Bjorn Poonen regarding questions that relate Mathematical Logic to Number Theory. 1. Our three “discussion problems.” Bjorn Poonen discussed the “recognition problem ” for finitely generated rings (and fields). That is, given tw ..."
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asked to take part in a questionandanswer session with Carol Wood and Bjorn Poonen regarding questions that relate Mathematical Logic to Number Theory. 1. Our three “discussion problems.” Bjorn Poonen discussed the “recognition problem ” for finitely generated rings (and fields). That is, given two finitely generated commutative rings A and B, presented in terms of generators and relations, is there a decision procedure to determine whether or not these rings are isomorphic (this being, one would think, a basic issue for algebraic geometry!). Of course, if one drops the requirement of commutativity, one comes up against the unsolvablity of the corresponding problem for finitely generated groups (by taking A and B simply to be integral group rings). Carol Wood brought up cases where model theory, applied to number theoretic problems provided bounds that are impressively good! Model Theory—in some instances—yields significantly new proofs of theorems obtained by the numbertheorists 2. In other instances, model theory achieves startling results for problems not yet considered by number theorists 3. Carol Wood discussed the recent article of Pila and Wilkie ([PW07]) that provides asymptotic upper bounds (as a function of the variable T) for the number of Qrational points of height ≤ T that 1
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.
ON DIOPHANTINE SETS OVER POLYNOMIAL RINGS
, 1999
"... We prove that the recursively enumerable relations over a polynomial ring R[t], where R is the ring of integers in a totally real number field, are exactly the Diophantine relations over R[t]. ..."
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We prove that the recursively enumerable relations over a polynomial ring R[t], where R is the ring of integers in a totally real number field, are exactly the Diophantine relations over R[t].