View source for Congruence with several variables

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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|Congruence modulo a prime number]]). A stronger result in this question has been obtained by P. Deligne [[#References|[3]]].

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

====Comments====
See also [[Congruence equation|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 $.