Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well-formed tensor formulas with explicit tensor entries is shown complete for \PhiP, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitz' theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts \PhiLOGCFL and \PhiL, and several other counting classes. Finally, the known inclusions NP=poly ` \PhiP=poly, LOGCFL=poly ` \PhiLOGCFL=poly, and NL=poly ` \PhiL=poly, which have scattered proofs in the literature [21, 39], are shown to follow from the new characterizations in a single blow. 1 Introduction Consider an algebraic structure S with certain operations. The following problem is sometimes called the word problem of S: given a reasonable encoding of a well-formed expression T ove...

Abstract. As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between 2M points for some integer M. The role of matrices is now played by 2M-dimensional “hypercubic ” arrays, and the determinant is replaced by a suitable generalization of it to such arrays – Cayley’s first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization. 1.

Abstract. The hypersurface amoeba membership problem is the problem to decide whether for an input point λ ∈ R n, the point λ is an element of the amoeba of the hypersurface generated by a multivariate complex Laurent polynomial c. We present an algorithm to solve this problem based on semidefinite programming and on Lasserre’s moment method for approximating the minimum of a real polynomial. The problem can be approximated by a sequence of semidefinite programs. In particular, we show that if the polynomial c is linear, this approach leads to a precise solution to the problem. 1.

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