Consider the given system of polynomial equations, where all the coefficients are in <mi mathvari

Jonathan Miles 2022-07-02 Answered
Consider the given system of polynomial equations, where all the coefficients are in C:
{ y n = P ( x ) Q ( x , y ) = 0
I would like to establish that either this system has solutions ( x , y ) C for all x C, or either has solutions for countably (or even better, finitely) many x C. I am not completely sure this is true but haven't found any counterexamples yet.
So far, I have found that if we have solutions for infinitely many x, then we have solutions for infinitely many y.
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)

lywiau63
Answered 2022-07-03 Author has 13 answers
You can determine the resultant R ( x ) = R e s y ( y n P ( x ) , Q ( x , y ) ) of both polynomials eliminating y. If it is the zero polynomial, then the polynomials have a common factor, which means that for any x you find at least one y so that ( x , y ) a solution (of the common factor and thus of the system).
If the resultant is not zero, then R ( x ) is a polynomial. Only at the roots of this polynomial you will then find at least one y so that ( x , y ) is a solution of the system.
Not exactly what you’re looking for?
Ask My Question

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 2020-12-03
2x+3y=1
−2x+3y=−7
asked 2022-05-23
Is the following equation regarded as a linear equation?
0 x 1 + 0 x 2 + 0 x 3 = 5
The original question is as below:
Solve the linear system given by the following augmented matrix:
( 2 2 3 1 2 5 3 0 0 0 0 5 )
Note the words linear system in the original question. So, I was asking myself whether 0 x 1 + 0 x 2 + 0 x 3 = 5 is a linear equation. Can we call all of the equations given by the matrix collectively as a linear system?
asked 2022-05-16
wo identitical pendula each of length l and with bobs of mass m are free to oscillate in the same plane. The bobs are joined by a spring with spring constant k, by looking for solutions where x and y vary harmonically at the same angular frequency ω, form a simultaneous equation for the amplitudes of oscillation x 0 and y 0 .
Considering the forces acting on each pendulum we can derive the following coupled-differential equations:
(1) m x ¨ = k ( y x ) m g x (2) m y ¨ = k ( y x ) m g y
Where x and y are the displacements of each of the pendulum as functions of time. If we assume they oscillate harmonically with angular frequency ω then we can write ω , ϕ 1 , ϕ 2 R:
x ( t ) = x 0 cos ( ω t + ϕ 1 ) y ( t ) = y 0 cos ( ω t + ϕ 2 )
Substituting these solutions back into ( 1 ) and ( 2 ) we get:
m ω 2 x 0 cos ( ω t + ϕ 1 ) = k ( x 0 cos ( ω t + ϕ 1 ) y 0 cos ( ω t + ϕ 2 ) ) m g cos ( ω t + ϕ 1 ) m ω 2 y 0 cos ( ω t + ϕ 2 ) = k ( x 0 cos ( ω t + ϕ 1 ) y 0 cos ( ω t + ϕ 2 ) ) m g cos ( ω t + ϕ 2 )
However, without assuming that ϕ 1 = ϕ 2 , in which case everything factors out nicely to leave a simultaneous equation in x 0 and y 0 , I cannot see a way of making it linear in x 0 and y 0 . So am I expected to use this assumption or is there a mathematical way of simplifying it?
If it is the former, then what would the physical justification for this assumption be?
asked 2022-05-21
How to solve coupled linear 1st order PDE
It is fairly straight forward to solve linear 1st order PDEs by the method of characteristics. For example, if
t f + a x f = b f
we have that d f d t = b f on the characteristic curve of d x d t = a. From this we deduce that f ( t , x ) = g ( C ) e b t where x = a t + C.
Now, how does this work when f is multidimensional. Can I solve equations on the following form by characteristics, or by any other means?
t f i ( t , x ) + j A i j x f j ( t , x ) = j B i j f j ( t , x )
where the components of A and B might be dependent on x and t.
In particular, I am trying to solve the following,
{ t f + c t x g = ( a + 1 t ) f t g + c t x f = ( b + 1 t ) g
where f and g are functions of x and t , where t > t 0 > 0, c 0. Any help is highly appreciated.
asked 2022-02-13
What is the solution to the system of equations below?
2x+3y=17
3x+6y=30
asked 2022-06-13
How to solve this system of equations
4 x + 2 y + 4 z = 0
2 x + y + 2 z = 0
4 x + 2 y + 4 z = 0
asked 2022-04-26
Solve the system of equations by elimination.
2a+5b=18
5a3b=14