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A third-order three-dimensional symmetric traceless tensor, called the octupolar tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar potential, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima capturing the most probable molecular orientations (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with three maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a separatrix surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible intra-octupolar transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction. Some other properties of octupolar tensors are also studied.

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order threedimensional symmetric and traceless tensor has four invariants with degrees two, four, six and ten respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no polynomial syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.Key words. minimal integrity basis, irreducible function basis, symmetric and traceless tensor, syzygy. Nomenclature D a third order three-dimensional symmetric and traceless tensor with components D ijk T(m, n) the space of real tensors of order m and dimension n S(m, n) the subspace of symmetric tensors St(m, n) the subspace of symmetric and traceless tensors O(n) the orthogonal group of dimension n SO(n) the special orthogonal group of dimension n Gl(n, R) the general linear group of real matrices m n = m! n!(m−n)! the binomial coefficient for m ≥ n ≥ 0The next theorem claims that there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 , where {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of D.Theorem 4.1. Let {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of a third order three-dimensional symmetric and traceless tensor D. Then there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 .