A Note on Cyclic Gradients

The cyclic derivative was introduced by G.-C. Rota, B. Sagan and P. R. Stein in [3] as an extension of the derivative to noncommutative polynomials. Here we show that there are simple necessary and sufficient conditions for an n-tuple of polynomials in n noncommuting indeterminates to be a cyclic gradient (see Theorem 1) and similarly for a polynomial to have vanishing cyclic gradient (see Theorem 2). Our interest in cyclic gradients stems from free probability theory and random matrices (see the Remark at the end) [1],[2],[4],[5],[6]. This note should also reduce the paucity of results on cyclic derivatives in several variables pointed out in [3, page 73].