Prove that for any two distinct points of an irreducible curve there exists a rational function that

Savanah Boone 2022-07-01 Answered
Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other.
I think I can construct such a function, for example, u ( x , y ) = ( x a ) 2 + ( y b ) 2 for given two points ( a , b ) and ( c , d ). However, this doesn't work for general algebraically closed field, for example, the case of ( c , d ) = ( a + i , b + 1 ). Hence now I have no clue. Could you give me a hint for this problem?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Zachery Conway
Answered 2022-07-02 Author has 7 answers
Perhaps this is a bit late, but here's what I first thought:
Suppose a c , b d, and characteristic is not 2. Then let u ( x , y ) := x a 2 ( c a ) + y b 2 ( d b ) . Then clearly u ( a , b ) = 0 and u ( c , d ) = 1 2 + 1 2 = 1

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-02-25
True or False. The graph of a rational function may intersect a horizontal asymptote.
asked 2022-09-12
How do you find the zeros of the function f ( x ) = 3 x 2 - 18 x + 24 x - 6 ?
asked 2022-07-04
I want to approximate tanh by a low-degree rational function of form
r ( x ) = p ( x ) q ( x ) = p 2 x 2 + p 1 x + p 0 q 1 x + q 0
such that the L 2 norm over the fixed interval [ x 0 , x 1 ] is small:
I ( p , q ) = 1 2 x 0 x 1 ( r ( x ) tanh ( x ) ) 2 d x .
That is, I would like to solve for the coefficients p 2 , p 1 , p 0 , q 1 , q 0 . Note that d e g ( p ) 2 and d e g ( q ) 1.
I don't know where to begin, so I'm looking for suggestions on how to approach this problem; pointers to relevant numerical methods would also be greatly appreciated
asked 2021-08-15
Find a function that has a vertical asymptote at x = - 7 and a horizontal asymptote at y = 5.
asked 2022-05-15
Express the following rational function in continued-fraction form:
4 x 2 + 3 x 7 2 x 3 + x 2 x + 5
The answer is :
(inline continued fraction) 4 2 x 1 2 + 23 8 x 63 92 406 529 x + 33 23
which means
4 2 x 1 2 + 23 8 x 63 92 406 529 x + 33 23
asked 2022-05-21
Could someone please let me know if my steps are correct? I am trying to rewrite the first line as a rational function, p ( x ) q ( x )
3 x 1 + 3 x 2 + 9 x 3 + 9 x 4 + 27 x 5 + 27 x 6 + = 3 x ( 1 + x ) + 9 x 3 ( 1 + x ) + 27 x 5 ( 1 + x ) + = 3 x ( 1 + x ) ( 1 + 3 x 2 + 9 x 4 + ) = ( 3 x + 3 x 2 ) 1 1 3 x 2 = 3 x + 3 x 2 1 3 x 2
asked 2022-06-20
We have recently started studying rational functions at school. I have learnt that a rational function is the ratio of two polynomials, i.e
p ( x ) q ( x )
My question is, what if p(x) and q(x) have a common factor (linear, quadratic, etc) ??
For example, ( x 1 ) ( x 2 ) ( x 2 ) ( x 3 )
Is it still a rational function? We haven’t studied calculus in maths yet, but we have studied a little bit of calculus in our physics classes, and from what I know, I can cancel out the common factor but the function is discontinuous at x = 2, i.e it has a hole.
So is it still a rational function?

New questions