Math Problem Inquiry Involving nonzero polynomials

[Ben1587]

Find a nonzero polynomial f(w, x, y, z) in the four indeterminates w, x, y, and z of minimum degree such that switching any two indeterminates in the polynomial gives the same polynomial except that its sign is reversed. For example, f(z, x, y,w) = -f(w, x, y, z). Prove that the degree of the polynomial is as small as possible.


No clue how to approach/sove.
Any advice/tips or solutions would be great!

thanks

 

[matt grime]

the best tip as always is for you to write out some polynomials ans swap over the variables and see what happens and try and find a solution that way

 

[M98Ranger]

To solve this problem, we first need to understand what a nonzero polynomial is. A nonzero polynomial is a polynomial with at least one nonzero coefficient. In other words, it is a polynomial that is not equal to zero for any value of its variables.

Now, let's consider the given conditions of the problem. We are looking for a polynomial in four variables, w, x, y, and z, of minimum degree such that switching any two variables results in the same polynomial with the sign reversed. This means that if we switch any two variables, say w and x, the resulting polynomial will be -f(x, w, y, z).

To approach this problem, we can start by considering polynomials of degree one. However, it is easy to see that a polynomial of degree one cannot satisfy the given conditions. For example, f(w, x, y, z) = w + x + y + z does not satisfy the condition that switching any two variables results in the same polynomial with the sign reversed.

Next, let's consider polynomials of degree two. A polynomial of degree two can be written as f(w, x, y, z) = aw^2 + bx^2 + cy^2 + dz^2 + ewx + fwy + gyz + hxz + ixz + jxy, where a, b, c, d, e, f, g, h, i, and j are coefficients. Now, let's switch the variables w and x and see if the resulting polynomial satisfies the given conditions.

f(x, w, y, z) = ax^2 + bw^2 + cy^2 + dz^2 + ewx + fwy + gyz + hxz + ixz + jxy

We can see that the polynomial is not the same as f(w, x, y, z) but it is also not the same as -f(w, x, y, z). Therefore, a polynomial of degree two also cannot satisfy the given conditions.

Next, let's consider polynomials of degree three. A polynomial of degree three can be written as f(w, x, y, z) = aw^3 + bx^3 + cy^3 + dz^3 + ew^2x + fw^2y + gw^2z + hx^2w + ix^2z + jx^2y + ky^2w + ly^2z + my^