Prove or disprove the following conjecture: 'For any n×n matrix A with real entries where each element a_ij = f(i,j) for some polynomial f(x,y) of degree at most 1, the determinant of A is 0 if and only if n ≥ 3 and the polynomial can be written as f(x,y) = g(x) + h(y) for some univariate polynomials g and h.' Your proof should: 1) State necessary definitions and mathematical concepts, 2) Establish lemmas that support your main argument, 3) Provide a rigorous proof of the statement, or a counterexample if disproving, 4) Discuss the implications of this theorem in relation to linear algebra and polynomial functions, 5) Consider generalizations or related theorems, 6) Explain the intuition behind the result in accessible terms.