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 indefinite 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 variables, etc.
The origins of these inequalities
The study of the Diophantine equations is 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. The arithmetic of Diophantus (ca. 3rd century AD) is a significant and main source that contains various types and systems of equations.
In this book, Diophantus foresaw a number of methods for studying the inequalities of the second and thirddegrees that were fully developed in the 19th century. The creation of the theory of rational numbers by this researcher of ancient Greece led to the analysis of logical solutions to indefinite systems, which are systematically followed in his book. Although his work contains solutions to 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). Based on this, conclusions were drawn that there is no single algorithm that could be used to determine for any given x whether the equation F (x, y1, …., y ). The situation is resolvable for y1, …, y . Examples of such polynomials can be written.
The simplest inequality
ax + by=1, where a and b are relatively integer and prime numbers, it has a huge number of executions (if x0, y0 the result is formed, then the pair of variables x=x0 + b and y=y0 -an, where n is arbitrary, will also be considered as an inequality). Another example of Diophantine equations is x2 + y2 =z2. The positive integral solutions of this inequality are the lengths of the small sides x, y and right triangles, as well as the hypotenuse z with integer side dimensions. These numbers are known as Pythagorean numbers. All triplets with respect to prime indicatedabove variables are given by x=m2 – n2, y=2mn, z=m2 + n2, where m and n are integers and prime numbers (m>n>0).
Diophantus in his "Arithmetic" is looking for rational (not necessarily integral) solutions of special types of his inequalities. A general theory for solving diophantine equations of the first degree was developed by C. G. Baschet in the 17th century. Other scientists at the beginning of the 19th century mainly studied similar inequalities like ax2 +bxy + cy2 + 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. Gauss for quadratic forms developed a general theory underlying some types of solutions.
In the study of these second-degree inequalities, significant progress was made only in the 20th century. A. Thue found that the Diophantine equation a0x + a1xn- 1y +…+a y =c, where n≧3, a0, …, a , c are integers, and a0tn + … + a cannot have an infinite number of integer solutions. However, Thue's method was not properly developed. A. Baker created effective theorems that give estimates on the performance of some equations of this kind. BN Delaunay proposed another method of investigation applicable to a narrower class of these inequalities. In particular, the form ax3 + y3 =1 is completely resolvable in this way.
Diophantine equations: methods of solution
The theory of Diophantus has many directions. Thus, a well-known problem in this system is the hypothesis that there is no non-trivial solution of the Diophantine equations xn + y =z n if n ≧ 3 (Fermat's question). The study of integer fulfillments of the inequality is a natural generalization of the problem of Pythagorean triplets. Euler obtained a positive solution of Fermat's problem for n=4. By virtue of this result, it refers to the proof of the missing integer, non-zero studies of the equation if n is an odd prime number.
The study regarding the decision has not been completed. The difficulties with its implementation are related to the fact that the simple factorization in the ring of algebraic integers is not unique. The theory of divisors in this system for many classes of prime exponents n makes it possible to confirm the validity of Fermat's theorem. Thus, the linear Diophantine equation with two unknowns is fulfilled by the existing methods and ways.
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 fulfilling inequalities of the form N(a1 x1 +…+ a x)=m, where N(a) is the norm of a, and x1, …, xn the integral rational variables are found. This class includes the Pell equation x2–dy2=1.
The values a1, …, a that appear, these equations are divided into two types. The first type - the so-called complete forms - include equations in which among a there are m linearly independent numbers over the field of rational variables Q, where m=[Q(a1, …, a):Q], in which there is a degree of algebraic exponents Q (a1, …, a ) over Q. Incomplete species are those in which the maximum number of a i less than m.
Full forms are simpler, their study is complete, 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) is an irreducible, homogeneous polynomial of degree n≧3. Thus, we can assume that yi→∞. Accordingly, if yi is large enough, then the inequality will contradict the theorem of Thue, Siegel and Roth, from which it follows that F(x, y)=C, where F is a form of the third degree or above, the irreducible cannot have an infinite number of solutions.
How to solve a Diophantine equation?
This example is a rather narrow class among all. For example, despite their simplicity, x3 + y3 + z3=N, and x 2 +y 2 +z2 +u2 =N are not included in this class. The study of solutions is a rather carefully studied branch of Diophantine equations, where the basis is the representation by quadratic forms of numbers. Lagrangecreated a theorem that says that the fulfillment exists for all natural N. Any natural number can be represented as the sum of three squares (Gauss's theorem), but it should not be of the form 4a(8K- 1), where a and k are non-negative integer exponents.
Rational or integral solutions to a system of a Diophantine equation of type F (x1, …, x)=a, where F (x 1, …, x) is a quadratic form with integer coefficients. Thus, according to the Minkowski-Hasse theorem, the inequality ∑aijxixj=b ij and b is rational, has an integral solution in real and p-adic numbers for every prime number p only if it is solvable in this structure.
Due to the inherent difficulties, the study of numbers with arbitrary forms of the third degree and above has been studied to a lesser extent. The main execution method is the method of trigonometric sums. In this case, the number of solutions to the equation is written explicitly in terms of the Fourier integral. After that, the environment method is used to express the number of fulfillment of the 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 samespheres. In the second half of the 19th century, the emergence of this number theory led to the study of the Diophantine 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 algebraic functions under study over a finite field of constants are one variable. Concepts such as class field theory, divisor, and 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 systematic research not only with numerical coefficients, but also with coefficients that 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, which arise as two equally important aspects 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 notion of a 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. One can accordingly say that the fundamental problem of Diophantine geometry is the study of rational pointsof an algebraic set X(K), while X are certain numbers in the field K. Integer execution has a geometric meaning in linear Diophantine equations.
Inequality studies and execution options
When studying rational (or integral) points on algebraic varieties, the first problem arises, which is their existence. Hilbert's tenth problem 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 definition of 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 Kv over all possible estimates. Since X(K)=X(Kv) are a necessary condition for existence, and the K point takes into account that the set X(Kv) is not empty for all v.
The importance lies in the fact that it brings together two problems. The second one is much simpler, it is solvable by a known algorithm. In the particular case where the variety X is projective, Hansel's lemma and its generalizations make further reduction possible: 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 more effective methods.
Lastan important consideration is that the sets X(Kv) are non-empty for all but a finite number of v, so the number of conditions is always finite and they can be effectively tested. However, Hasse's principle does not apply to degree curves. For example, 3x3 + 4y3=5 has points in all p-adic number fields and in system of real numbers, but has no rational points.
This method served as a starting point for constructing a concept describing the classes of principal homogeneous spaces of Abelian varieties to perform a "deviation" from the Hasse principle. It is described in terms of a special structure that can be associated with each manifold (Tate-Shafarevich group). The main difficulty of the theory lies in the fact that methods for calculating groups are difficult to obtain. This concept has also been extended to other classes of algebraic varieties.
Search for an algorithm for fulfilling inequalities
Another heuristic idea used in the study of Diophantine equations is that if the number of variables involved in a set of inequalities is large, then the system usually has a solution. However, this is very difficult to prove for any particular case. The general approach to problems of this type uses analytic number theory and is based on estimates for trigonometric sums. This method was originally applied to special kinds of equations.
However, later it was proved with its help that if the form of an odd degree is F, in dand n variables and with rational coefficients, then n is large enough compared to d, so the projective hypersurface F=0 has a rational point. According to Artin's conjecture, this result is true even if n > d2. This has only been proven for quadratic forms. Similar problems can be asked for other fields as well. The central problem of Diophantine geometry is the structure of the set of integer or rational points and their study, and the first question to be clarified is whether this set is finite. In this problem, the situation usually has a finite number of executions if the degree of the system is much larger than the number of variables. This is the basic 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 of whether these numbers are bounded on the set of all elliptic curves over a given field K has been studied. The boundedness of the torsion 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 conjecture states that the number of integral points is finite for a curve of genus g>1. In the functional case, this concept was demonstrated by Yu. I. Manin in 1963. The main tool used in proving finiteness theorems in Diophantine geometry is the height. Of the algebraic varieties, dimensions above one are abelianmanifolds, which are the multidimensional analogs of elliptic curves, have been the most thoroughly studied.
A. Weil generalized the theorem on the finiteness of the number of generators of a group of rational points to Abelian varieties of any dimension (the Mordell-Weil concept), extending it. In the 1960s, the conjecture of Birch and Swinnerton-Dyer appeared, improving this and the group and the zeta functions of the manifold. Numerical evidence supports this hypothesis.
Solvability problem
The problem of finding an algorithm that can be used to determine whether any Diophantine equation has a solution. An essential feature of the problem posed 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.p1=0, …, PK=0p=0, …, pK=0 or p21+ ⋯ + P2K=0. n12+⋯+pK2=0. The problem of finding such a universal way to find solutions for linear inequalities in integers was posed by D. Gilbert.
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 conjecture appeared, which said that any enumerable set also belongs to the Greek scientist. Because examples of algorithmically undecidable sets are known, but are recursively enumerable. It follows that the Davis conjecture is true and the problem of solvability of these equationshas a negative execution.
After that, for the Davis conjecture, it remains to prove that there is a method for transforming an inequality that also (or did not) at the same time have a solution. It was shown that such a change of the Diophantine equation is possible if it has the above two properties: 1) in any solution of this type v ≦ uu; 2) for any k, there is an execution with exponential growth.
An example of a linear Diophantine equation of this class completed the proof. The problem of the existence of an algorithm for the solvability and recognition of these inequalities in rational numbers is still considered an important and open question that has not been studied sufficiently.
