Algebraic inequalities or their systems with rational coefficients whose solutions are sought in integral or integer numbers. As a rule, the number of unknowns in diophantine equations is greater. Thus, they are also known as vague inequalities. In modern mathematics, the above concept is applied to algebraic equations whose solutions are sought in algebraic integers of some extension of the field of Q-rational variables, the field of p-adic ones, etc.
The origins of these inequalities
The study of Diophantus equations lies on the border between number theory and algebraic geometry. Finding solutions in integer variables is one of the oldest mathematical problems. Already at the beginning of the second millennium BC the ancient Babylonians managed to solve systems of equations with two unknowns. This branch of mathematics flourished most in ancient Greece. Arithmetic of Diophantus (approximately, 3rd century AD) is a significant and main source, which contains various types and systems of equations.
In this book, Diophantus foresaw a number of methods for studying inequalities of the second and third degrees, which were fully developed in the 19th century. The creation of the theory of rational numbers by this researcher of Ancient Greece led to an analysis of the logical solutions of indefinite systems, which are systematically accompanied in his book. Despite the fact that his work contains solutions of specific Diophantine equations, there is reason to believe that he was also familiar with several general methods.
The study of these inequalities is usually associated with serious difficulties. Due to the fact that they contain polynomials with integer coefficients F (x, y1, ..., y n ). Based on this, it was concluded that there is no single algorithm with which it would be possible to determine for any given x whether the equation F (x, y 1 , ..., y n ) is satisfied. The situation is decidable for y 1 , ..., y n . Examples of such polynomials can be written.
Simple inequality
ax + by = 1, where a and b are relatively integers and primes, there are a huge number of executions for it (if x 0, y 0 the result is generated, then a pair of variables x = x 0 + b n and y = y 0 -an , where n is arbitrary, will also be considered as the fulfillment of the inequality). Another example of diophantine equations is x 2 + y 2 = z 2 . The positive integral solutions of this inequality are the lengths of the small sides x, y and right triangles, as well as the hypotenuses z with integer lateral dimensions. These numbers are known as Pythagorean numbers. All triplets with respect to the simple variables indicated above are given by the formulas x = m 2 - n 2 , y = 2mn, z = m 2 + n 2 , where m and n are integers and primes (m> n> 0).
Diophantus in his "Arithmetic" is engaged in the search for rational (not necessarily integral) solutions of special types of his inequalities. The general theory of solving Diophantine equations of the first degree was developed by K. G. Bashet in the 17th century. Other scientists at the beginning of the 19th century mainly studied similar inequalities like ax 2 + bxy + cy 2 + dx + ey + f = 0, where a, b, c, d, e, and f are general, heterogeneous, with two unknowns of the second degree . Lagrange used continued fractions in his study. For quadratic forms, Gauss developed the general theory underlying the solution of certain types.
In studies of these inequalities of the second degree, significant successes were achieved only in the 20th century. A. Thue found that the Diophantine equation a 0 x n + a 1 x n-1 y + ... + a n y n = c, where n≥3, a 0 , ..., a n , c are integers, and a 0 t n + ... + a n cannot have an infinite number of integer solutions. However, the Thue method was not properly developed. A. Baker created effective theorems that give estimates on the fulfillment of certain equations of this kind. B. N. Delone proposed another research method applicable to a narrower class of these inequalities. In particular, the form ax 3 + y 3 = 1 is completely solvable in this way.
Diophantine equations: solution methods
Diophantus theory has many directions. Thus, the well-known problem in this system is the hypothesis according to which there is no non-trivial solution of the Diophantine equations x n + y n = z n if n ≥ 3 (Fermat's question). The study of integer inequalities is a natural generalization of the problem of Pythagorean triplets. Euler obtained a positive solution to the Fermat problem for n = 4. By virtue of this result, it relates to the proof of missing integer, nonzero studies of the equation if n is an odd prime.
The study regarding the solution was not completed. Difficulties with its implementation are connected with the fact that simple factorization in a ring of algebraic integers is not unique. The divisor theory in this system for many classes of simple exponents n allows us to confirm the validity of Fermat's theorem. Thus, using the existing methods and methods, a linear Diophantine equation with two unknowns is satisfied.
Types and types of described tasks
Arithmetic of rings of algebraic integers is also used in many other problems and solutions of Diophantine equations. For example, such methods were applied when satisfying inequalities of the form N (a 1 x 1 + ... + a n x n ) = m, where N (a) is the norm of a, and integral rational variables were found x, 1 , ..., x n . This class includes the Pell equation x 2– dy 2 = 1.
The values of a 1, ..., a n that appear, these equations are divided into two types. The first type - the so-called full forms - includes equations in which among a there are m linearly independent numbers over the field of rational variables Q, where m = [Q (a 1 , ..., a n ): Q], in which there is a degree algebraic exponents Q (a1, ..., a n ) over Q. Incomplete species are those in which the maximum number a i is less than m.
Complete forms are simpler, their research is completed, and all solutions can be described. The second type - incomplete species - is more complicated, and the development of such a theory has not yet been completed. Such equations are studied using Diophantine approximations, which include the inequality F (x, y) = C, where F (x, y) - a polynomial of degree n≥3 is irreducible, homogeneous. Thus, we can assume that y i → ∞. Accordingly, if y i is large enough, then the inequality will contradict the Thue, Siegel, and Roth theorem, from which it follows that F (x, y) = C, where F is a form of third degree or higher, an irreducible one cannot have an infinite number of solutions.
How to solve the Diophantine equation?
This example is a fairly narrow class among all. For example, despite their simplicity, x 3 + y 3 + z 3 = N, and also x 2 + y 2 + z 2 + u 2 = N are not included in this class. The study of solutions is a fairly thoroughly investigated branch of Diophantine equations, where the basis is represented by quadratic forms of numbers. Lagrange created a theorem which states that fulfillment exists for all natural N. Any natural number can be represented as the sum of three squares (Gauss theorem), but it should not have the form 4 a (8K-1), where a and k are non-negative whole indicators.
Rational or integral solutions of a system of a diophantine equation of type F (x 1 , ..., x n ) = a, where F (x 1 , ..., x n ) is a quadratic form with integer coefficients. Thus, according to the Minkowski-Hasse theorem, the inequality ∑a ij x i x j = b where a ij and b is rational, has an integral solution in real and p-adic numbers for each prime p only when it is solvable in this structure .
Due to the inherent difficulties, the study of numbers with arbitrary forms of the third degree and higher has been less studied. The main implementation method is the method of trigonometric sums. In this case, the number of solutions of the equation is explicitly written in terms of the Fourier integral. After that, the environment method is used to express the amount of inequality of the corresponding congruences. The method of trigonometric sums depends on the algebraic features of the inequalities. There are a large number of elementary methods for solving linear Diophantine equations.
Diophantine analysis
Department of Mathematics, the subject of which is the study of integral and rational solutions of systems of equations of algebra by methods of geometry, from the same sphere. In the second half of the 19th century, the appearance of this number theory led to the study of Diophantus equations from an arbitrary field with coefficients, and solutions were considered either in it or in its rings. The system of algebraic functions developed in parallel with numbers. The basic analogy between the two, which was emphasized by D. Hilbert and, in particular, L. Kronecker, led to the uniform construction of various arithmetic concepts, which are usually called global.
This is especially noticeable if the studied algebraic functions over a finite field of constants are one variable. Concepts such as class field theory, a divisor, and also branching and results are a good illustration of the above. This point of view was adopted in the system of Diophantine inequalities only later, and a systematic study not only with numerical, but also with coefficients, which are functions, began only in the 1950s. One of the decisive factors in this approach was the development of algebraic geometry. The simultaneous study of the fields of numbers and functions that arise as two equally important sides of the same subject not only gave elegant and convincing results, but led to the mutual enrichment of the two topics.
In algebraic geometry, the concept of variety is replaced by a non-invariant set of inequalities over a given field K, and their solutions are replaced by rational points with values in K or in its finite extension. We can say, accordingly, that the fundamental problem of Diophantine geometry is to study the rational points of the algebraic set X (K), while X is certain numbers in the field K. Integer execution has geometric meaning in linear Diophantine equations.
Inequality Studies and Options
When studying rational (or integral) points on algebraic varieties, the first problem arises, which consists in their existence. Hilbert's tenth task is formulated as the problem of finding a general method for solving this problem. In the process of creating an exact definition of the algorithm and after it was proved that there are no such executions for a large number of problems, the problem acquired an obvious negative result, and the most interesting question is the determination of the classes of Diophantine equations for which the above system exists. The most natural approach, from an algebraic point of view, is the so-called Hasse principle: the initial field K is studied together with its completions K v according to all possible estimates. Since X (K) = X (K v ) are a necessary condition for existence, and the K point takes into account that the set X (K v ) is not empty for all v.
The importance is that it brings two problems together. The second is much simpler; it is solvable by a well-known algorithm. In the particular case when the variety X is projective, the Hansel lemma and its generalizations make possible further reduction: the problem can be reduced to the study of rational points over a finite field. Then he decides to build a concept either through consistent research, or by more effective methods.
The last important consideration is that the sets X (K v ) are nonempty for all v except for a finite number, so the number of conditions is always finite, and they can be tested effectively. However, the Hasse principle is not applicable to degree curves. For example, 3x 3 + 4y 3 = 5 has points in all p-adic number fields and in the system of real numbers, but does not have rational points.
This method served as a starting point for constructing a concept that describes the classes of principal homogeneous spaces of Abelian varieties for performing a “deviation” from the Hasse principle. It is described in terms of a special structure that can be associated with each variety (Tate-Shafarevich group). The main difficulty of the theory is that methods for calculating groups are difficult to obtain. This concept has also been extended to other classes of algebraic varieties.
Search for an inequality fulfillment algorithm
Another heuristic idea used in the study of Diophantine equations is that if the number of variables involved in the set of inequalities is large, then the system usually has a solution. However, it is very difficult to prove for any particular case. A general approach to problems of this type uses analytical number theory and is based on estimates of trigonometric sums. This method was originally applied to special types of equations.
However, it was subsequently proved with its help that if the form of an odd degree is F, in d and n variables and with rational coefficients, then n is large enough in comparison with d, so the projective hypersurface F = 0 has a rational point. According to the hypothesis Artin, this result is true even if n> d 2 . This is proved only for quadratic forms. Similar problems can be posed for other fields. The central problem of Diophantine geometry is the structure of the set of integer or rational points and their study, and the first question that needs to be clarified is whether this set is finite. In this problem, a situation usually has a finite number of executions if the degree of the system is much greater than the number of variables. This is the main assumption.
Inequalities on lines and curves
The group X (K) can be represented as a direct sum of a free structure of rank r and a finite group of order n. Since the 1930s, the question has been studied whether these numbers are bounded on the set of all elliptic curves over a given field K. The torsion boundedness n was demonstrated in the seventies. There are curves of arbitrary high rank in the functional case. In the numerical case, there is still no answer to this question.
Finally, Mordell’s hypothesis states that the number of integral points is finite for a curve of genus g> 1. In a functional case, this concept was demonstrated by Yu. I. Manin in 1963. The main tool used in proving finiteness theorems in Diophantine geometry is height. Of algebraic varieties of dimension higher than one, Abelian varieties, which are multidimensional analogues of elliptic curves, have been most thoroughly studied.
A. Weil generalized the theorem on the finiteness of the number of generators of a group of rational points on Abelian varieties of any dimension (Mordell-Weil concept), extending it. In the 1960s, the Birch and Swinnerton-Dyer hypotheses appeared that improved this group and the zeta functions of manifolds. Numerical evidence supports this hypothesis.
Solvability problem
The problem of finding an algorithm with which you can determine whether any Diophantine equation has a solution. An essential feature of the task is the search for a universal method that would be suitable for any inequality. Such a method would also allow solving the above systems, since it is equivalent to P21 + ⋯ + P2k = 0.n1 = 0, ..., PK = 0n = 0, ..., nK = 0 or n21 + ⋯ + P2K = 0. n12 + ⋯ + nK2 = 0. The problem of finding such a universal method for finding solutions for linear inequalities in integers was posed by D. Hilbert.
In the early 1950s, the first studies appeared aimed at proving the non-existence of an algorithm for solving Diophantine equations. At this time, the Davis hypothesis appeared, which stated that any enumerated set also belongs to the Greek scientist. Since examples of algorithmically unsolvable sets are known, but are recursively enumerable. It follows that the Davis hypothesis is true and the solvability problem for these equations has a negative fulfillment.
After this, for the Davis hypothesis, it remains to prove that there is a method of transforming inequality that also (or did not have) at the same time a solution. It was shown that such a change in the Diophantine equation is possible if it has the indicated two properties: 1) in any solution of this type v ≤ uu ; 2) for any k, there is a fulfillment in which there is exponential growth.
An example of a linear diophantine equation of this class completed the proof. The problem of the existence of a solvability and recognition algorithm in rational numbers of these inequalities is still considered an important and open question, which has not been studied sufficiently.