A congruence of the form
where
is a polynomial in the variables
with integral rational coefficients
and
is an integer. The maximum value of the magnitude
where the maximum is taken over all possible tuples
for which
(
),
is called the
degree with respect to the set of variables
or the
degree
of
(1).
The maximum value of the magnitudes
,
,
where the maximum is taken over the same tuples
,
is called the
degree of the congruence equation with respect to the variable
.
The principal problem in the theory of congruence equations is the
number of solutions of a given congruence. It is possible to restrict
the problem to the case of a prime module, since the
problem of the number of solutions of
(1)
by a composite module
,
except for a few degenerate cases, may be reduced to
the problem of the number of solutions of the congruences
(
)
for the prime modules
that are divisors of
.
The most thoroughly studied congruence equations
(
)
in one variable are the two-term congruences (cf.
Two-term congruence)
The study of the number of solutions of congruences in the case of a general polynomial
is very difficult, and only isolated partial solutions have so far been obtained.
A system of congruences
may be considered as a system of algebraic equations:
over the finite prime field
consisting of
elements; the number of solutions of this system of
congruences will be equal to the number of
-rational
points of the
algebraic variety
defined by the system of equations
(2).
Accordingly,
together with number-theoretical methods, the methods of algebraic
geometry are also used in the study of
congruence equations or systems of such congruences.
The most extensively studied congruence equations in
several variables were those of the form
The estimate
was obtained for the number of solutions
of a congruence of this type, where
is an absolutely irreducible polynomial. The constant
depends only on the polynomial and is equal to the genus of the curve
.
Such an estimate for the first non-trivial case, viz. for the elliptic congruence
was obtained by
H. Hasse
in
1934,
who based his work
on the formula for the addition of points on the
Jacobi variety
of the curve
.
Hasse's method was subsequently extended by
A. Weil
[4]
to the case of absolutely irreducible polynomials
.
This estimate was also obtained in
[3]
by elementary methods.
Studies of congruence equations with a number of variables
have been much less extensive. One general result is the
theorem of Chevalley,
according to which if
is a form the degree of which is strictly smaller than the
number of variables, then the number of solutions of the congruence
is positive and is divisible by
;
in the case of non-homogeneous polynomials the existence
of a solution is not guaranteed, but the divisibility by
remains valid
(Warning's theorem).
This last theorem has been generalized to systems of congruences.
The theory of congruence equations has numerous applications in
other branches of number theory — the
theory of Diophantine equations, problems in additive
number theory, algebraic number theory, etc.