- #1
swampwiz
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Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be:
For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots?
I was looking at this explanation @ Youtube, which explains that if the polynomial variable is some value at some very large magnitude, the value of the function as per that value of the polynomial variable is a loop that must circle the origin with the number of times - i.e., the winding number - being the degree of the polynomial, and then as this magnitude is modulated continuous to smaller values, this loop will collapse onto the point that is the value of the free term of the polynomial (let's presume it is non-zero), with the winds of the loops crossing the origin, thus signifying that there exists a root with the magnitude of that root being the modulated radius of the circle, and such that further modulation of the radius to lower values will result in that wind of the loop never again winding around, thus reducing the winding number by 1. And so finally, since the loop that is simply the degenerate value of the free term has a winding number of 0, there must be the number of roots that is equal to the degree of the polynomial.
OK, so if the roots are all at different magnitudes, I can see this happening. However, if there a multiple roots at the same magnitude - and even more so if there are multiple roots (i.e., same magnitude and complex angle) - I am confused about how this schema handles this situation.
For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots?
I was looking at this explanation @ Youtube, which explains that if the polynomial variable is some value at some very large magnitude, the value of the function as per that value of the polynomial variable is a loop that must circle the origin with the number of times - i.e., the winding number - being the degree of the polynomial, and then as this magnitude is modulated continuous to smaller values, this loop will collapse onto the point that is the value of the free term of the polynomial (let's presume it is non-zero), with the winds of the loops crossing the origin, thus signifying that there exists a root with the magnitude of that root being the modulated radius of the circle, and such that further modulation of the radius to lower values will result in that wind of the loop never again winding around, thus reducing the winding number by 1. And so finally, since the loop that is simply the degenerate value of the free term has a winding number of 0, there must be the number of roots that is equal to the degree of the polynomial.
OK, so if the roots are all at different magnitudes, I can see this happening. However, if there a multiple roots at the same magnitude - and even more so if there are multiple roots (i.e., same magnitude and complex angle) - I am confused about how this schema handles this situation.