Let F(x1,...,xn) be a polynomial with real coefficients in n>1 “indeterminate” variables x1,...,xn. We say that F is n-alternating if for all integers 1≤i<j≤n, F(x1,...,xi,...,xj,...,xn)=−F(x1,...,xj,...,xi,...,xn), i.e. swapping the order of indeterminates xi,xj flips the sign of the polynomial. For example, x12x2−x22x1 is 2-alternating, whereas x1x2x3+2x2x3 is not 3-alternating.Note: two polynomials P(x1,...,xn) and Q(x1,...,xn) are considered equal if and only if each monomial constituent ax1k1...xnkn of P appears in Q with the same coefficient a, and vice versa. This is equivalent to saying that P(x1,...,xn)=0 if and only if every possible monomial constituent of P has coefficient 0.(1) Compute a 3-alternating polynomial of degree 3.
(2) Prove that the degree of any nonzero n-alternating polynomial is at least (2n).