Forms in many variables
Lead Research Organisation:
Royal Holloway University of London
Department Name: Mathematics
Abstract
A Diophantine equation is an equation to be solved in integers (the sequence 1,2,3,... are the positive integers), for example(*) x^2+y^2=z^2has the integer solution (x,y,z)=(3,4,5). This example is a so called quadratic equation ,since all terms occur as a square. One can also consider Diophantine inequalities, for which oneis interested in making a certain quantity, for example a x^2 + b y^2 + c z^2 very small , where now a,b,c are real numbers like 3.141..., but x,y,y are still integers.The proposed research deals with an important subclass of Diophantine equations/Diophantine inequalitiesand aims to advance our knowledge in several key aspects, in particular:1) Since dealing with individual Diophantine problems is extremely hard, we aim to do better on average .2) Our second goal is to study quantitative aspects like the number of solutions for quadratic Diophantine equations.3) We want to discuss not only solutions of Diophantine equations in integers, but also in primes numbers(a prime number is an integer exceeding 1 and only divisible by 1 and itself, like 2,3,5,...)This research addresses key aspects of pure mathematics, in particular number theory, and its main benefitsand applications are are again in pure mathematics, in particular number theory.
Planned Impact
This research proposal concerns research in pure mathematics, and its main impact will be again on pure mathematics. The proposed research is not only relevant in its original domain, namely Analytic Number Theory, but also to neighboring fields like modular and quadratic forms or Diophantine Approximation. As usual for research in pure mathematics, this does not preclude future applications to other fields outside mathematics, or applications of economic interest, but these effects are very difficult to predict and usually work on a very long term basis, counting in decades rather than in years. Pure mathematics is an essential part of our culture, and advancing our knowledge in pure mathematics therefore also advances our culture. Moreover, this research proposal includes a one year position for a postdoctoral research assistant, this way contributing to the training of the next generation of UK researchers.
People |
ORCID iD |
| Rainer Dietmann (Principal Investigator) |
Publications
Brüdern J
(2012)
Random Diophantine inequalities of additive type
in Advances in Mathematics
Brüdern J
(2014)
Random Diophantine equations, I
in Advances in Mathematics
Brüdern J
(2012)
Random diophantine equations, I
Dietmann R
(2013)
On the representation of quadratic forms by quadratic forms
in Michigan Mathematical Journal
Dietmann R
(2012)
Weyl's inequality and systems of forms
Dietmann R
(2013)
Probabilistic Galois theory
in Bulletin of the London Mathematical Society
Dietmann R
(2011)
Probabilistic Galois Theory
Dietmann R
(2013)
On the representation of quadratic forms by quadratic forms
Dietmann R
(2014)
WEYL'S INEQUALITY AND SYSTEMS OF FORMS
in The Quarterly Journal of Mathematics
Dietmann R
(2012)
Hilbert cubes in progression-free sets and in the set of squares II
| Description | A Diophantine equation is an equation to be solved in integers (the sequence 1,2,3,... are the positive integers), for example (*) x^2+y^2=z^2 has the integer solution (x,y,z)=(3,4,5). This example is a so called "quadratic equation", since all terms occur as a square. One can also consider Diophantine inequalities, for which one is interested in making a certain quantity, for example a x^2 + b y^2 + c z^2 "very small", where now a,b,c are real numbers like 3.141..., but x,y,y are still integers. The proposed research deals with several important subclasses of Diophantine equations/Diophantine inequalities and aims to advance our knowledge in several key aspects, in particular: 1) Since dealing with individual Diophantine problems is extremely hard, we aim to do better "on average", i.e. establish results that hold "for almost all" equations. 2) Our second goal is to study quantitative aspects like the number of solutions for certain quadratic Diophantine equations. 3) To equations in one variable such as x^3+2013*x^2-1972*x+1991 one can attach a certain group of symmetries, the so called Galois group. One expects that most of these equations have as many symmetries as possible. We want to establish a good quantitative form of this belief. Addressing mainly these three questions, but also a few others, we in particular obtained the following outcomes in this research project: 1) For the first question, we found that as soon as s is at least 2d+1, for "almost all" real numbers a_1, ..., a_s the Diophantine inequality |a_1 x_1^d + ... + a_s x_s^d| < e for any given positive e has infinitely many integer solution x_1, ..., x_s. In fact, one can even asymptotically count the number of solutions. This appears to be the first such "metric" result for this problem. We obtained a similar result, that is slightly more complicated to state, for Diophantine equations instead of Diophantine inequalities. 2) For the second question, we could show that if F is an integer symmetric positive definite s by s matrix, B is an integer symmetric positive definite integer m by m matrix, and s is sufficiently large in terms of m and some geometric aspect of B, then for the number of integral s by m matrices A that are solutions of the Diophantine equation A^t F A = B, the expected asymptotic formula holds true. This appears to be the first result of this form assuming only a very mild geometric condition on B. 3) Regarding the third question, we could show that the number of monic integer polynomials X^n + a_1 X^{n-1} + ... + a_n with coefficients of size at most H, that do not admit as many symmetries as possible (i.e. "their Galois group is smaller than S_n"), can be bounded by a constant times H^{n-0.58...}. This improves the previous world record H^{n-0.5} from 1973. We also dealt with a few other questions, for example we could show that if p is a prime, then for any integer n in the interval [1, p] one can change at most 0.11... r binary digits of n in order to arrive at a primitive root modulo p, where r denotes the number of binary digits of p. This goes beyond what a simple application of Burgess classical results on gaps between primitive roots would yield. |
| Exploitation Route | By using ideas, techniques or results (=theorems) from our papers in order to prove new theorems. |
| Sectors | Digital/Communication/Information Technologies (including Software) |
| Description | Our findings have inspired both us and also other mathematicians to continue research in this area, but no use as suggested in this section has materialized so far. Given the nature of pure mathematics, this is certainly not surprising. |