# Congruence with several variables

A congruence

$$\tag{1 } f( x _ {1} \dots x _ {n} ) \equiv 0 ( \mathop{\rm mod} m),$$

where $f( x _ {1} \dots x _ {n} )$ is a polynomial in $n \geq 2$ variables with integer rational coefficients not all of which are divisible by $m$. The solvability of this congruence modulo $m = p _ {1} ^ {\alpha _ {1} } {} \dots p _ {s} ^ {\alpha _ {s} }$, where $p _ {1} \dots p _ {s}$ are different prime numbers, is equivalent to the solvability of the congruences

$$\tag{2 } f( x _ {1} \dots x _ {n} ) \equiv \ 0 ( \mathop{\rm mod} p _ {i} ^ {\alpha _ {i} } )$$

for all $i = 1 \dots s$. The number $N$ of solutions of (1) is then equal to the product $N _ {1} \dots N _ {s}$, where $N _ {i}$ 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

$$\tag{3 } f( x _ {1} \dots x _ {n} ) \equiv 0 ( \mathop{\rm mod} p ^ \alpha ),\ \ \alpha > 1,$$

to be solvable, it is necessary that the congruence

$$\tag{4 } f( x _ {1} \dots x _ {n} ) \equiv 0 ( \mathop{\rm mod} p)$$

modulo a prime number $p$ 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 $x _ {i} \equiv x _ {i} ^ {(} 1)$( $\mathop{\rm mod} p$) of (4) such that $( df/dx _ {i} )( x _ {1} ^ {(} 1) \dots x _ {n} ^ {(} 1) ) \not\equiv 0$( $\mathop{\rm mod} p$) for at least one $i = 1 \dots n$, generates $p ^ {( \alpha - 1)( n- 1) }$ solutions $x _ {i} \equiv x _ {i} ^ {( \alpha ) }$( $\mathop{\rm mod} p ^ \alpha$) of (3), whereby $x _ {i} ^ {( \alpha ) } \equiv x _ {i} ^ {(} 1)$( $\mathop{\rm mod} p$) when $i = 1 \dots n$.

Thus, in the non-degenerate case, the question of the number of solutions of the congruence (1) modulo a composite number $m$ reduces to the question of the number of solutions of congruences of the form (4) modulo the prime numbers $p$ that divide $m$. If $f( x _ {1} \dots x _ {n} )$ is an absolutely-irreducible polynomial with integer rational coefficients, then for the number $N _ {p}$ of solutions of (4), the estimate

$$| N _ {p} - p ^ {n-} 1 | \leq C( f ) p ^ {n-} 1- 1/2$$

holds, where the constant $C( f )$ depends only on $f$ and does not depend on $p$. It follows from this estimate that the congruence (4) is solvable for all prime numbers $p$ that are larger than a certain effectively-calculable constant $C _ {0} ( f )$, depending on the given polynomial $f( x _ {1} \dots x _ {n} )$( 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

See also Congruence equation for more information. A polynomial $f ( x _ {1} \dots x _ {n} )$ over $\mathbf Q$ is absolutely irreducible if it is still irreducible over any (algebraic) field extension of $\mathbf Q$.