# Reduction of the coefficients of a polynomial in sage

Given the two polynomials in two variables x and y

```
A(y,x)=((a1)*y+(a2))*((a3)*x+(a4))
B(y,x)=((b4)-(b3)*x)*((b2)-(b1)*y)
```

both congruent to zero mod a semiprimal number N

Is there a method in sage to reduce , the coefficients in x and y smaller than sqrt (N) and the coefficient in xy smaller than 64, of their linear combination of the same degree of polynomials A and B by exploiting the congruence?

Example

given the two polynomials in two variables x and y

```
A(y,x)=(25*y+11)*(27*x+1)
B(y,x)=(65-8*x)*(67-8*y)
```

both congruent to zero mod 1763

I had thought of finding m and n such that

```
m*(a1)*(a3)+n*(b1)*(b3) = N*t +T
m*(a1)*(a4)-n*(b1)*(b4) = N*s + S
m*(a2)*(a3)-n*(b2)*(b3) = N*w + W
T <= 64
S <= sqrt(N)
W <= sqrt(N)
```