Positivity, real-rootedness and log-concavity

For a finite sequence a0 through ad, log-concavity means ai squared is at least ai−1 ai+1 at every interior index. Assume the entries are nonnegative and there are no internal zeros, meaning the positive indices form a single interval. Then successive positive ratios are nonincreasing, so the sequence can change from rising to falling at most once; it is unimodal, allowing a plateau. Without the support assumption, the locally checked inequalities can miss a long gap between separated positive entries.

Nonnegative coefficients alone imply neither log-concavity nor real-rootedness. Even a positive log-concave sequence can have nonreal roots: the polynomial 1+t+t² has coefficients (1,1,1) satisfying equality in its interior adjacent-product comparison, but discriminant minus three. Conversely, a polynomial with mixed-sign coefficients may have only real roots without giving the desired nonnegative counting sequence. Root statements and coefficient-shape statements must therefore be compared within an explicitly declared nonnegative class.

Suppose A(t) has degree d, nonnegative coefficients and only real roots. A positive real argument cannot be a root because all summands are nonnegative and at least one is positive. Hence all nonzero roots are negative; any zero roots contribute an initial power of t. Factoring the remaining polynomial into positive multiples of terms t+r with r>0 shows that its positive coefficients occupy a consecutive interval. Thus the support condition needed for unimodality follows from these hypotheses.

Newton inequalities assert log-concavity after dividing coefficient ai by the binomial coefficient choose(d,i). This is stronger than ordinary log-concavity because the binomial coefficients themselves are strictly log-concave in the interior. One route identifies normalized coefficients with normalized elementary symmetric functions of nonnegative root parameters. We use the theorem rather than prove its full inequality here. Use the actual degree d for this sharp normalization; padding the coefficient list with trailing zeros changes the binomial factors and must not be confused with the same normalized sequence.

Littlewood–Richardson coefficients are nonnegative because they count tableaux or representation multiplicities. That does not force every polynomial formed from them to have real roots. Likewise a Cohen–Macaulay complex has a nonnegative h-vector, but this alone provides no universal unimodality or real-rootedness theorem. The triangle boundary already has h(t)=1+t+t² and nonreal roots. Geometry or representation theory can supply stronger conditions in particular families, but each conclusion needs the relevant additional theorem.

Positivity also changes under a change of basis. For example, p2=m2 is monomial-positive, yet p2=s2−s11 is not Schur-positive. Schur-positivity does imply monomial-positivity because Kostka coefficients are nonnegative. Reversing that implication fails in this example. A productive research question therefore names the objects, the coefficient basis or generating statistic, and the exact strength of the desired property before looking for a combinatorial interpretation or a real-rootedness-preserving recurrence.