A tensor has border rank at most r if it can be written as T=lim_ε→ 0 T(ε) where T(ε) has rank at most r for all sufficiently small ε. It is known that the map ε↦ T(ε) can be assumed to be a (tensor valued) polynomial in ε. The smallest possible degree of such a map is called the error degree of T. The error degree and the related notion of order of degeneration are the two key quantities that we study in this paper. One motivation comes from debordering: by polynomial interpolation on the map ε↦ T(ε) we can upper bound the tensor rank of T. For order 3 tensors, exponential upper bounds on the error degree and degeneration order were given almost 40 years ago in (Lehmkuhl Lickteig, 1989) and were not improved ever since. In this paper we give bounds that apply to a wide class of tensors, exponentially improving on (Lehmkuhl Lickteig, 1989). Our results are most general for tensors with 3 slices (format m × n × 3). In this case, our main assumption is on the rank of the matrix slices. We also give bounds that apply to arbitrary rectangular formats (m × n × p). In this case, we need an additional 1-regularity assumption on one of the slices of the tensor (recall that a matrix is said to be 1-regular if its eigenspaces are 1-dimensional). Under these assumptions we show that the error degree is at most 1, which yields a nontrivial debordering result (tensor rank at most 2r for border rank r). The results in (Lehmkuhl Lickteig, 1989) rely on an upper bound on the degree of the variety of tensors of border rank at most r. We rely instead on more specific properties of this algebraic variety, and in particular on commutativity properties of certain matrices derived from the tensor slices.

We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank). As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices.

We give a new, constructive uniqueness theorem for tensor decomposition. It applies to order 3 tensors of format n × n × p and can prove uniqueness of decomposition for generic tensors up to rank r=4n/3 as soon as p ≥ 4. One major advantage over Kruskal's uniqueness theorem is that our theorem has an algorithmic proof, and the resulting algorithm is efficient. Like the uniqueness theorem, it applies in the range n ≤ r ≤ 4n/3. As a result, we obtain the first efficient algorithm for overcomplete decomposition of generic tensors of order 3. For instance, prior to this work it was not known how to efficiently decompose generic tensors of format n × n × n and rank r=1.01n (or rank r ≤ (1+ϵ) n, for some constant ϵ >0). Efficient overcomplete decomposition of generic tensors of format n × n × 3 remains an open problem. Our results are based on the method of commuting extensions pioneered by Strassen for the proof of his 3n/2 lower bound on tensor rank and border rank. In particular, we rely on an algorithm for the computation of commuting extensions recently proposed in a companion paper, and on the classical diagonalization-based "Jennrich algorithm" for undercomplete tensor decomposition.

In a recent paper, we studied a modified version of the Einstein-Rosen bridge. This modified bridge is traversable and works as a one-way membrane: a particle on the first sheet falling toward the throat will reach it in finite time (in Eddington coordinates), and will continue its trajectory on the second sheet. In this paper, we show that the particle undergoes a PT-symmetry as it crosses the throat. This could lead to observable effects thanks to an additional ingredient proposed by Einstein and Rosen: congruent points on the two sheets are identified. We propose a bimetric model to realize this identification for our modified bridge.

We study symmetric tensor decompositions, i.e., decompositions of the form $T = \sum_{i=1}^r u_i^{\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \in \mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent. This implies $r \leq n$,that is, the decomposition of T is undercomplete. We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercomplete decomposition.Then given some $T'$ close to $T$, an accuracy parameter $\varepsilon$, and an upper bound B on the condition number of the tensor, output vectors $u'_i$ such that $||u_i - u'_i|| \leq \varepsilon$ (up to permutation and multiplication by cube roots of unity) with high probability. The main novel features of our algorithm are: 1) We provide the first algorithm for this problem that runs in linear time in the size of the input tensor. More specifically, it requires $O(n^3)$ arithmetic operations for all accuracy parameters $\varepsilon =$ 1/poly(n) and B = poly(n). 2) Our algorithm is robust, that is, it can handle inverse-quasi-polynomial noise (in $n$,B,$\frac{1}{\varepsilon}$) in the input tensor. 3) We present a smoothed analysis of the condition number of the tensor decomposition problem. This guarantees that the condition number is low with high probability and further shows that our algorithm runs in linear time, except for some rare badly conditioned inputs. Our main algorithm is a reduction to the complete case ($r=n$) treated in our previous work [Koiran,Saha,CIAC 2023]. For efficiency reasons we cannot use this algorithm as a blackbox. Instead, we show that it can be run on an implicitly represented tensor obtained from the input tensor by a change of basis.

A tuple (Z_1,...,Z_p) of matrices of size r is said to be a commuting extension of a tuple (A_1,...,A_p) of matrices of size n <r if the Z_i pairwise commute and each A_i sits in the upper left corner of a block decomposition of Z_i. This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

We study symmetric tensor decompositions, i.e. decompositions of the input symmetric tensor T of order 3 as sum of r 3rd-order tensor powers of u_i where u_i are vectors in \C^n. In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from the u_i. In this paper we assume that the u_i are linearly independent. This implies that r is at most n, i.e., the decomposition of T is undercomplete. We will moreover assume that r=n (we plan to extend this work to the case where r is strictly less than n in a forthcoming paper). We give a randomized algorithm for the following problem: given T, an accuracy parameter epsilon, and an upper bound B on the condition number of the tensor, output vectors u'_i such that u_i and u'_i differ by at most epsilon (in the l_2 norm and up to permutation and multiplication by phases) with high probability. The main novel features of our algorithm are: (1) We provide the first algorithm for this problem that works in the computation model of finite arithmetic and requires only poly-logarithmic (in n, B and 1/epsilon) many bits of precision. (2) Moreover, this is also the first algorithm that runs in linear time in the size of the input tensor. It requires O(n^3) arithmetic operations for all accuracy parameters epsilon = 1/poly(n).

We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomised algorithm for the following problem: If a homogeneous polynomial $$f \in K[x_1,..., x_n]$$ (where $$K \subseteq \mathbb {C}$$ ) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are: These results are true even when we replace $$\mathbb {C}$$ by $$\mathbb {R}$$ . We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to overcomplete decompositions, i.e. decompositions in more than n powers of linear forms.

We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial $f \in K[x_1 , . . . , x_n]$ (where $K \subseteq \mathbb{C}$) of degree $d$ is given as a blackbox, decide whether it can be written as a linear combination of $d$-th powers of linearly independent complex linear forms. The main novel features of the algorithm are: (1) For $d = 3$, we improve by a factor of $n$ on the running time from an algorithm by Koiran and Skomra. The price to be paid for this improvement though is that the algorithm now has two-sided error. (2) For $d > 3$, we provide the first randomized blackbox algorithm for this problem that runs in time polynomial in $n$ and $d$ (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorization subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms. (3) For $d > 3$, when $f$ has rational coefficients, the running time of the blackbox algorithm is polynomial in $n,d$ and the maximal bit size of any coefficient of $f$. This yields the first algorithm for this problem over $\mathbb{C}$ with polynomial running time in the bit model of computation.

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 sum 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.

The Eddington-Finkelstein metric is obtained from the Schwarzschild metric by a change of the time variable. It is well known that a test mass falling into a black hole does not reach the event horizon for any finite value of the Schwarzschild time variable t. By contrast, we show that the event horizon is reached for a finite value of the Eddington-Finkelstein time variable t'. Then we study in Eddington-Finkelstein time the fate of a massive particle traversing an Einstein-Rosen bridge and obtain a different conclusion than recent proposals in the literature: we show that the particle reaches the wormhole throat for a finite value t'_1 of the time marker t', and continues its trajectory across the throat for t'>t'_1. Such a behavior does not make sense in Schwarzschild time since it would amount to continuing the trajectory of the particle "beyond the end of time."

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 some 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).

We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ℓ^n where ℓ is the lacunary size of the input polynomial and n its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in ℓ and n. Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any. Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.

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 Strassen's theorem fits within this framework we also provide a self-contained proof of his lower bound.

Let F(x,y)∈C[x,y] be a polynomial of degree d and let G(x,y)∈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 that the multiplicity of any isolated solution (a,b)∈C2 with nonzero coordinates is no greater than 52d2t2. 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.

We give a separation bound for the complex roots of a trinomial f∈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.

A sum of affine powers is an expression of the form [f(x 1 ,...,x n ) = ∑ i =1 s α i l i (x 1 ,...,x n ) e i ] where l i is an affine form. We propose polynomial time black-box algorithms that find the decomposition with the smallest value of s for an input polynomial f . Our algorithms work in situations where s is small enough compared to the number of variables or to the exponents e i . Although quite simple, this model is a generalization of Waring decomposition. This paper extends previous work on Waring decomposition as well as our work on univariate sums of affine powers (ISSAC'17).

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 lower 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.

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