# Multivariate quartic function help

• rattam
In summary, the conversation discusses a difficult question involving the equation xy=1 and the solution x^4+y^4-z^4+4z^2-2=0. The student has tried various methods but has not been able to solve it and is seeking help. The solution may involve a line on the xy=1 plane in 3D and a closed form equation.
rattam

## Homework Statement

I am a Turkish student bogged down by a very difficult question (at least for me).

Here's the question:

Given that xy= 1, find all solutions for X4 + y4 - z4 + 4z2 - 2 = 0.

I've spent many hours on this equation. It even prevents me from sleep.

I would appreciate it deeply if someone were to solve it for me, but I'll be equally grateful being directed to a book where equations of this kind are handled. Is this a trivial question? Is it simple to answer with the necessary background?

## Homework Equations

the multiplication of xy is constant and equal to 1.

## The Attempt at a Solution

I have tried various ways of expanding it, tried to rewrite the z part as binomial, but I think this is just beyond my skill.

Thanks so much for pputting up with my desperation! Any response is appreciated very much.

Last edited:
I would say the solution is some kind of a line on the xy=1 plane in 3D. Your solution would be a closed form of an equation. So I don't know if saying that
{(x,1/x,z) satisfying the equation x^4+1/x^4-z^4-4z^2-2=0 }
is the solution set is enough.

## What is a multivariate quartic function?

A multivariate quartic function is a mathematical function that involves multiple variables and has a degree of four. It is represented by an equation in the form of f(x,y,z) = ax^4 + bx^3y + cx^2y^2 + dx^2yz + ey^4 + fxyz + gx^2z^2 + hxz^3 + iz^4.

## What is the purpose of a multivariate quartic function?

A multivariate quartic function can be used to model complex relationships between multiple variables. It is often used in fields such as physics, economics, and engineering to analyze and predict outcomes.

## How do you graph a multivariate quartic function?

To graph a multivariate quartic function, you will need to plot points on a three-dimensional coordinate system. This can be done by assigning values to each variable and calculating the corresponding output. The resulting points can then be connected to form a graph.

## What is the difference between a univariate and a multivariate quartic function?

A univariate quartic function has only one variable, while a multivariate quartic function has multiple variables. This means that a univariate quartic function can only be graphed on a two-dimensional coordinate system, while a multivariate quartic function requires a three-dimensional coordinate system.

## What are some real-life applications of multivariate quartic functions?

Multivariate quartic functions can be used to model and analyze a wide range of phenomena, such as the trajectory of a projectile, the behavior of chemical reactions, and the relationship between supply and demand in economics. They are also used in machine learning and data analysis to identify patterns and make predictions.

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