Problem 6

An ordered pair

(x,y)

of integers is called a primitive point if

\gcd(x,y)=1

. Given a finite set

S

of primitive points, prove that there exist a positive integer

n

and integers

a_0,a_1,\ldots,a_n

such that for every

(x,y)\in S

, we have

a_0x^n+a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}+a_ny^n=1

Step 3 of 8: Record the determinant corrections

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In plain words

The determinant vanishes at an old point and will be used to preserve all old values.

d_i=y_ix_k-x_iy_k

Detailed analysis

For each old point $(x_i,y_i)$ , define $d_i=y_ix_k-x_iy_k$ , and put $D=\prod_{i=1}^{k-1}d_i$ . By Bezout applied to the new primitive point, choose integers a,b with $ax_k+by_k=1$ . We will add a correction term that vanishes at every old point but has a controlled value at the new one.