Does the (piecewise straight border) line of a polygon have an equation, as a circle has the equation x²+y²=1?

Yes, but I don't know why you'd want to use it. We'll approach the answer in steps, by examples.

1. Here is an infinite line: x+y-1=0.
2. Here is an alternate form: (x+y-1)²=0 Note that the left hand side cannot be negative.
3. Here is an equation of a curve that satisfies either of two conditions A(x,y)²=0 or B(x,y)²=0:
   A(x,y)² B(x,y)² = 0
4. Here is an equation of a curve that satisfies both conditions A(x,y)²=0 and B(x,y)²=0:
   A(x,y)² + B(x,y)² = 0
5. Here's how to turn the inequality x>=0 into an equation. q is a variable that is not used anywhere else.
   q² = x
6. Here is a finite segment, or edge, of the above line, for 0<=x<=10:
   (x+y-1)² + (q²-(25-(x-5)²)) = 0
7. Now you have enough to find the equation for a whole polygon.

The reason that the above works is that these are complex curves that have small regions that are real.