0 like 0 dislike
0 like 0 dislike
There exists soluction by radicals for some class of multivariate complex polynomials?

3 Answers

0 like 0 dislike
0 like 0 dislike
Am I missing anything?
For any Y,Z,W, just take
X = -(Y + Y\*Z + 10\*Y\*W)/10
and this is a solution (and pretty clearly all of them) yes?
0 like 0 dislike
0 like 0 dislike
Reduce modulo 10 and you find y(1-z)=0 (mod 10). That gives 4 cases for (y,z) modulo 10. Plugging y=10k gives x=k(10w+z-1). So take any positive k, any positive w, any positive z, and this equation gives you x. For example (w,x,y,z)=(1, 10, 10, 1) is a solution. More generally, (w,k(10w+z-1),10k,z) is a solution for any positive w,k,z. I assume the other 4 cases are equally challenging.
by
0 like 0 dislike
0 like 0 dislike
It’s worth pointing out that even without a change of variables you can solve the original equation by noticing that it is homogeneous in x and y.  This means that if (x,y,z,w) is a solution then so is (dx,dy,z,w) for any d.  In particular, if you divide through by the gcd(x,y) you may assume that gcd(x,y) = 1.

But then notice that in the equation 10x + y - yz - 10yw = 0 you have that y divides 10x.  But if gcd(x,y) = 1 then y divides 10 and so y = 1,2,5 or 10.  You can easily solve the original equation by plugging in the various values of y and then solving for z.

For example if y = 5 then

10x + 5 - 5z - 50w = 0 yields z = 2x - 10w + 1.  So you are free to use any x,w and then you have the solutions (dx, 5d, 2x-10w+1, w) where d,x,w are arbitrary positive integers and gcd(5,x) = 1.

You can do this for each value of y to parametrize all solutions to the original equation.

Related questions

0 like 0 dislike
0 like 0 dislike
61 answers
_spunkki asked Jun 21
Just ordered a Klein Bottle from Cliff Stoll. He sent me about 2 dozen pictures of him packing it up. Why is he so cute :)
_spunkki asked Jun 21
0 like 0 dislike
0 like 0 dislike
5 answers
BrianDenver7 asked Jun 21
Is there a nice way to recast riemannian geometry in terms of principal bundles?
BrianDenver7 asked Jun 21
0 like 0 dislike
0 like 0 dislike
47 answers
countykathleen asked Jun 21
Is it possible to suck at rigor of math but great at intuition?
countykathleen asked Jun 21
0 like 0 dislike
0 like 0 dislike
3 answers
daedaePIVOT asked Jun 21
Cofactor Matrix of Cofactor Matrix
daedaePIVOT asked Jun 21
0 like 0 dislike
0 like 0 dislike
3 answers
jc_coleman4 asked Jun 21
Examples of distributions
jc_coleman4 asked Jun 21

33.4k questions

135k answers

0 comments

33.7k users

OhhAskMe is a math solving hub where high school and university students ask and answer loads of math questions, discuss the latest in math, and share their knowledge. It’s 100% free!