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 1 of 8: Induction begins with Bezout

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In plain words

A single primitive point admits an integer linear form taking value 1.

P(x,y)=ax+by

Detailed analysis

We induct on the size of $S$ . For the base case, write the only point as $(x_1,y_1)$ . Primitivity gives integers a and b with $ax_1+by_1=1$ . Set $P(x,y)=ax+by$ ; it is homogeneous of positive degree and $P(x_1,y_1)=1$ .