Question 97
Given a positive integer $n \ge 3$. find the minimum integer $k$ satisfying the following property. For every $n$ Points $A_i = (x_i, y_i)$ on the Plane, no three of them being collinear, and for every real numbers $c_i$ ($1 \le i \le n$) there exists a polynomial $P(x, y)$ such that $P(x_i, y_i) = c_i$ for all $i = 1, \dots, n$, and the Degree of $P$ is not heater than $k$. (F. petrov) given a natural number $n \ge 3$. For which smallest $k$ is the following statement true? For any $n$ points $A_i = (x_i, y_i)$ on a plane, no three of which are on the same line, and any real numbers $c_i$ ($1 \le i \le n$) exists Such a polynomial $P(x, y)$, the degree of which is not greater than $k$, which $P(x_i, y_i) = c_i$ for all $i = 1, \dots, n$. (a polynomial from two variables is a function of the form $$
P(x, y) = a_{0,0} + a_{1,0}x + a_{0,1}y + a_{2,0}x^2 + a_{1,1}xy + a_{0,2}y^2 + \dots + a_{k,0}x^k + a_{k-1,1}x^{k-1}y + \dots + a_{0,k}y^k.
$$ by the degree of non-zero monomial $a_{i,j}x^i y^j$ is called the number $i+j$; the degree of the polynomial $P(x, y)$ The highest degree of the monomial included in it is called.
P(x, y) = a_{0,0} + a_{1,0}x + a_{0,1}y + a_{2,0}x^2 + a_{1,1}xy + a_{0,2}y^2 + \dots + a_{k,0}x^k + a_{k-1,1}x^{k-1}y + \dots + a_{0,k}y^k.
$$ by the degree of non-zero monomial $a_{i,j}x^i y^j$ is called the number $i+j$; the degree of the polynomial $P(x, y)$ The highest degree of the monomial included in it is called.