Abstract: In this paper, we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both the tensor-tensor multiplication and the tensor polynomial evaluation problem under this multiplication is the same as that of the matrix multiplication, unless the linear map is injective. As for non-algebraic functions, we define the tensor geometric mean and the tensor Wasserstein mean for pseudo-positive-definite tensors under the tensor-tensor multiplication with invertible linear maps, and we show that the tensor geometric mean can be calculated by solving a specific Riccati tensor equation. Furthermore, we show that the tensor geometric mean does not satisfy the resultantal (determinantal) identity in general, which the matrix geometric mean always satisfies. Then we define a pseudo-SVD for the injective linear map case, and we apply it to image and video data compression.

Abstract: In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. Specifically, we define the geometric mean of two T-positive definite tensors and verify several properties that "mean" should satisfy including the idempotence and the commutative property, and so on. Moreover, it is shown that the geometric mean is a unique T-positive definite solution of an algebraic Riccati tensor equation and can be expressed as solutions of algebraic Riccati matrix equations. In addition, we investigate the Riemannian manifold associated with the geometric mean for T-positive definite tensors, considering it as a totally geodesic embedded submanifold of the Riemannian manifold associated with the case of matrices. It is particularly shown that the geometric mean of two T-positive definite tensors is the midpoint of a unique geodesic joining the tensors, and the manifold is a Cartan-Hadamard-Riemannian manifold.

Abstract: We calculate the ordinary C₂-cohomology of BT² with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to those of BT¹ and BU(2), calculated previously, both relationships being more complicated than in the nonequivariant case.

Abstract: Computing the Voronoi diagram of mixed geometric objects in R³ is challenging due to the high cost of exact geometric predicates via Cylindrical Algebraic Decomposition (CAD). We propose an efficient exact verification framework that characterizes the parameter space connectivity by computing certified topological transition sets. We analyze the fundamental non-quadric case: the trisector of two skew lines and one circle in R³. Since the bisectors of circles and lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable. Our pipeline uses exact symbolic evaluations to identify transition walls. Jacobian computations certify the absence of affine singularities, while projective closure shows singular behavior is isolated at a single point at infinity, $p_{\infty}$. Tangent-cone analysis at  $p_{\infty}$ yields a discriminant $\Delta_Q = 4ks^2 (k-1)$, identifying k=0, 1 as bifurcation values. Using directional blow-up coordinates, we rigorously verify that the trisector's real topology remains locally constant between these walls. Finally, we certify that k=0, 1 are actual topological walls exhibiting reducible splitting. This work provides the exact predicates required for constructing mixed-object Voronoi diagrams beyond the quadric-only regime.

Abstract: In this article, we investigate the rank index of projective curves $C \subset P^r$ of degree r+1 when $C = \pi_{p} (\tilde{C})$ for the standard rational normal curve $\tilde{C} \subset P^{r+1}$ and a point $p \in P^{r+1} \setminus \tilde{C}^{3}$ where $\tilde{C}^k$ denotes the k-fold self-join of $\tilde{C}$. Here, the rank index of a closed subscheme $X \subset P^r$ is defined to be the least integer k such that the homogeneous ideal of X can be generated by quadratic polynomials of rank ≤ k. Our results show that the rank index of C is at most 4, and it is exactly equal to 3 when the projection center p is a coordinate point of $P^{r+1}$. We also investigate the case where $p \in \tilde{C}^3 \setminus \tilde{C}^2$.

Abstract: Grassmann cactus variety is a common generalisation of Grassmann secant variety and cactus variety. In their definitions one considers the vector spaces of fixed dimension that are contained in the linear span of some finite schemes. We prove that to characterise Grassmann cactus varieties it is enough to use finite schemes that locally have low socle dimension. This motivates the study of parameter spaces of such schemes and simplifies calculations of examples of Grassmann cactus varieties.

Abstract: We show that the border rank of the 4 × 4 determinant tensor is at least 12 over C, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly 12.

Abstract: We generalize the celebrated Fröberg's theorem to embedded joins of copies of a simplicial complex, namely higher secant complexes to the simplicial complex, in terms of property Nq+1,p due to Green and Lazarsfeld. Furthermore, we investigate combinatorial phenomena parallel to geometric ones observed for higher secant varieties of minimal degree.

Abstract: We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on R(detₙ) completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks R(det₄)=12 and R(perm₄)=8 over arbitrary field of characteristic ≠ 2.

Abstract: White's conjecture predicts quadratic generators for the ideal of any matroid base polytope. We prove that White's conjecture for any matroid M implies it also for any matroid M′, where M and M′ differ by one basis. Our study is motivated by inner projections of algebraic varieties.

Abstract: In this paper, we study minimal generators of the (saturated) defining ideal of $\sigma_k(v_d(\mathbb{P}^n))$ in $\mathbb{P}^N$ with $N=\binom{n+d}{d} -1$, the k-secant variety of d-uple Veronese embedding of projective n-space, of a relatively small degree. We first show that the prime ideal $I(\sigma_4(v_3(\mathbb{P}^3)))$ can be minimally generated by 36 homogeneous polynomials of degree 5. It implies that $\sigma_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$ is a del Pezzo 4-secant variety (i.e., $\deg(\sigma_4(v_3(\mathbb{P}^3))) = 105$ and the sectional genus $\pi(\sigma_4(v_3(\mathbb{P}^3))) = 316$) and provides a new example of an arithmetically Gorenstein variety of codimension 4. As an application, we decide non-singularity of a certain locus in $\sigma_4(v_3(\mathbb{P}^3))$. By inheritance, generators of $I(\sigma_4(v_3(\mathbb{P}^n)))$ are also obtained for any $n \geq 3$. We also propose a procedure to compute the first non-trivial degree piece $I(\sigma_k(v_d(\mathbb{P}^n)))_{k+1}$ for a general k-th secant case, in terms of prolongation and weight space decomposition, based on the method used for $\sigma_4(v_3(\mathbb{P}^3))$ and treat a few more cases of k-secant varieties of the Veronese embedding of a relatively small degree in the end.