Obtaining superlinear lower bounds on tensor rank is a major open problem in complexity theory. In this paper we propose a generalization of the approach used by Strassen in the proof of his 3n/2 border rank lower bound. Our approach revolves around
a problem on commuting matrices: Given matrices Z_1,...,Z_p of size n and an integer r>n, are there commuting matrices Z_1,...,Z_p of size r such that every Z_k is embedded as a submatrix in the top-left corner of Z_k? As one of our main results, we show that this question always has a positive answer for r larger than rank(T)+n, where T denotes the tensor with slices Z_1,..,Z_p. Taking the contrapositive, if one can show for some specific matrices Z_1,...,Z_p and a specific integer r that this question has a negative answer, this yields the lower bound rank(T) > r-n. There is a little bit of slack in the above rank(T)+n bound, but we also provide a number of exact characterizations of tensor rank and symmetric rank, for ordinary and symmetric tensors, over the fields of real and complex numbers. Each of these characterizations points to a corresponding variation on the above approach. In order to explain how Strassens theorem fits within this framework we also provide a self-contained proof of his lower bound.
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be written as a s
um of cubes of linearly independent linear forms with complex coefficients. Compared to previous algorithms for the same problem, the two main novel features of this algorithm are: (i) It is an algebraic algorithm, i.e., it performs only arithmetic operations and equality tests on the coefficients of the input polynomial. In particular, it does not make any appeal to polynomial factorization. (ii) For an input polynomial with rational coefficients, the algorithm runs in polynomial time when implemented in the bit model of computation. The algorithm relies on methods from linear and multilinear algebra (symmetric tensor decomposition by simultaneous diagonalization). We also give a version of our algorithm for decomposition over the field of real numbers. In this case, the algorithm performs arithmetic operations and comparisons on the input coefficients. Finally we give several related derandomization results on black box polynomial identity testing, the minimization of the number of variables in a polynomial, the computation of Lie algebras and factorization into products of linear forms.
We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new proof of som
e of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called ASD property).
Let $F(x, y) in mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y) in mathbb{C}[x,y]$ be a polynomial with $t$ monomials. We want to estimate the maximal multiplicity of a solution of the system $F(x,y) = G(x,y) = 0$. Our main result is th
at the multiplicity of any isolated solution $(a,b) in mathbb{C}^2$ with nonzero coordinates is no greater than $frac{5}{2}d^2t^2$. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of $F$ and $G$, and we briefly review some connections between sparse polynomials and algebraic complexity theory.
This paper is devoted to the factorization of multivariate polynomials into products of linear forms, a problem which has applications to differential algebra, to the resolution of systems of polynomial equations and to Waring decomposition (i.e., de
composition in sums of d-th powers of linear forms; this problem is also known as symmetric tensor decomposition). We provide three black box algorithms for this problem. Our main contribution is an algorithm motivated by the application to Waring decomposition. This algorithm reduces the corresponding factorization problem to simultaenous matrix diagonalization, a standard task in linear algebra. The algorithm relies on ideas from invariant theory, and more specifically on Lie algebras. Our second algorithm reconstructs a factorization from several bi-variate projections. Our third algorithm reconstructs it from the determination of the zero set of the input polynomial, which is a union of hyperplanes.
We give a separation bound for the complex roots of a trinomial $f in mathbb{Z}[X]$. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of $f$; in particular, it is polynomial in $log (deg f)$. It is
known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of $f$ rather than the number of monomials) give separation bounds that are exponentially worse.As an algorithmic application, we show that the number of real roots of a trinomial $f$ can be computed in time polynomial in the size of the sparse encoding of~$f$. The same problem is open for 4-nomials.
We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lo
wer bound on the dimension of the space of polynomials spanned by $F$. In particular, we give simple criteria ensuring that the dimension of the span of $F$ is at least $c.|F|$ for some absolute constant $c<1$. We also propose conjectures implying the linear independence of the elements of $F$. These conjectures are known to be true for the field of real numbers, but not for the field of complex numbers.
The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complex
ity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives? We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the trace method, recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.A slightly shorter version of this paper was presented at STACS17. In this new version we have corrected a typo in Section 4.1, and added a reference to Shitovs work on tensor rank.
In this paper we study sums of powers of affine functions in (mostly) one variable. Although quite simple, this model is a generalization of two well-studied models: Waring decomposition and sparsest shift. For these three models there are natural ex
tensions to several variables, but this paper is mostly focused on univariate polynomials. We present structural results which compare the expressive power of the three models; and we propose algorithms that find the smallest decomposition of f in the first model (sums of affine powers) for an input polynomial f given in dense representation. We also begin a study of the multivariate case. This work could be extended in several directions. In particular, just as for Sparsest Shift and Waring decomposition, one could consider extensions to supersparse polynomials and attempt a fuller study of the multi-variate case. We also point out that the basic univariate problem studied in the present paper is far from completely solved: our algorithms all rely on some assumptions for the exponents in an optimal decomposition, and some algorithms also rely on a distinctness assumption for the shifts. It would be very interesting to weaken these assumptions, or even to remove them entirely. Another related and poorly understood issue is that of the bit size of the constants appearing in an optimal decomposition: is it always polynomially related to the bit size of the input polynomial given in dense representation?
In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a represen
tation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order $Omega$($sqrt$ d), and were obtained from arguments based on Wronskian determinants and shifted derivatives. We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as lacunary polynomial interpolation).
