Congruence with several variables

A congruence

(1)

where is a polynomial in variables with integer rational coefficients not all of which are divisible by . The solvability of this congruence modulo , where are different prime numbers, is equivalent to the solvability of the congruences

(2)

for all . The number of solutions of (1) is then equal to the product , where is the number of solutions of (2). Thus, when studying congruences of the form (1) it is sufficient to confine oneself to moduli that are powers of prime numbers.

For a congruence

(3)

to be solvable, it is necessary that the congruence

(4)

modulo a prime number be solvable. In non-degenerate cases, the solvability of (4) is also a sufficient condition for the solvability of (3). More precisely, the following statement is correct: Every solution () of (4) such that () for at least one , generates solutions () of (3), whereby () when .

Thus, in the non-degenerate case, the question of the number of solutions of the congruence (1) modulo a composite number reduces to the question of the number of solutions of congruences of the form (4) modulo the prime numbers that divide . If is an absolutely-irreducible polynomial with integer rational coefficients, then for the number of solutions of (4), the estimate

holds, where the constant depends only on and does not depend on . It follows from this estimate that the congruence (4) is solvable for all prime numbers that are larger than a certain effectively-calculable constant , depending on the given polynomial (see also Congruence modulo a prime number). A stronger result in this question has been obtained by P. Deligne [3].

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2]	H. Hasse, "Zahlentheorie" , Akademie Verlag (1963)
[3]	P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–307

Comments

See also Congruence equation for more information. A polynomial over is absolutely irreducible if it is still irreducible over any (algebraic) field extension of .