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 2 of 8: Handle opposite points first

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In plain words

A homogeneous polynomial of even degree has the same value at opposite points.

P(x,y)^2

Detailed analysis

Assume the induction hypothesis gives a homogeneous integer polynomial $P(x,y)=ax+by$ of degree n equal to 1 on the existing k-1 points. Let the new primitive point be $q=(x_k,y_k)$ . If $q=-(x_i,y_i)$ for an old point, then $P^2$ still equals 1 on every old point and also at $q=(x_k,y_k)$ , by homogeneity. Thus we may assume $d_i=y_ix_k-x_iy_k$ is nonzero for every old point.