According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result and establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree. We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions: 1. We deborder this orbit closure and some related orbit closures, i.e., prove all points in the orbit closure have small non-border algebraic branching programs. 2. We fully implement the geometric complexity theory approach against the power sum by generalizing the ideas of Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, we obtain new multiplicity obstructions that are constructed from just the symmetries of the polynomials.

In the algebraic metacomplexity framework we prove that the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. We use this to resolve an open question posed by Grochow, Kumar, Saks & Saraf (2017). Our result means that many existing algebraic complexity lower bound proofs can be efficiently converted into isotypic lower bound proofs via highest weight metapolynomials, a notion studied in geometric complexity theory. In the context of algebraic natural proofs, it means that without loss of generality algebraic natural proofs can be assumed to be isotypic. Our proof is built on the Poincaré-Birkhoff-Witt theorem for Lie algebras and on Gelfand-Tsetlin theory, for which we give the necessary comprehensive background.

Valiant’s famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential. Informally, VNP is an exponential sum of VP-circuits. In this paper we study whether, in general, exponential sums (of algebraic circuits) require exponential-size algebraic circuits. We show that the famous Shub-Smale τ-conjecture indeed implies such an exponential lower bound for an exponential sum. Our main tools come from parameterized complexity. Along the way, we also prove an exponential fpt (fixed-parameter tractable) lower bound for the parameterized algebraic complexity class VW⁰_{nb}[𝖯], assuming the same conjecture. VW⁰_{nb}[𝖯] can be thought of as the weighted sums of (unbounded-degree) circuits, where only ± 1 constants are cost-free. To the best of our knowledge, this is the first time the Shub-Smale τ-conjecture has been applied to prove explicit exponential lower bounds. Furthermore, we prove that when this class is fpt, then a variant of the counting hierarchy, namely the linear counting hierarchy collapses. Moreover, if a certain type of parameterized exponential sums is fpt, then integers, as well as polynomials with coefficients being definable in the linear counting hierarchy have subpolynomial τ-complexity. Finally, we characterize a related class VW[𝖥], in terms of permanents, where we consider an exponential sum of algebraic formulas instead of circuits. We show that when we sum over cycle covers that have one long cycle and all other cycles have constant length, then the resulting family of polynomials is complete for VW[𝖥] on certain types of graphs.

We study algebraic complexity classes and their complete polynomials under \emph{homogeneous linear} projections, not just under the usual affine linear projections that were originally introduced by Valiant in 1979. These reductions are weaker yet more natural from a geometric complexity theory (GCT) standpoint, because the corresponding orbit closure formulations do not require the padding of polynomials. We give the \emph{first} complete polynomials for VF, the class of sequences of polynomials that admit small algebraic formulas, under homogeneous linear projections: The sum of the entries of the non-commutative elementary symmetric polynomial in 3 by 3 matrices of homogeneous linear forms. Even simpler variants of the elementary symmetric polynomial are hard for the topological closure of a large subclass of VF: the sum of the entries of the non-commutative elementary symmetric polynomial in 2 by 2 matrices of homogeneous linear forms, and homogeneous variants of the continuant polynomial (Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of circuits with arity-3 product gates.

Given an Abelian group 𝒢, a Boolean-valued function f: 𝒢 → {-1,+1}, is said to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the domain 𝒢. In a seminal paper, Gopalan et al. [Gopalan et al., 2011] proved "Granularity" for Fourier coefficients of Boolean valued functions over ℤ₂ⁿ, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over ℤ₂ⁿ which are approximately Fourier-sparse. In this work, we obtain structural results for approximately Fourier-sparse Boolean valued functions over Abelian groups 𝒢 of the form, 𝒢: = ℤ_{p_1}^{n_1} × ⋯ × ℤ_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m²s)^⌈φ(m)/2⌉, on the absolute value of the smallest non-zero Fourier coefficient of an s-sparse function, where m = p_1 ⋯ p_t, and φ(m) = (p_1-1) ⋯ (p_t-1). We carefully apply probabilistic techniques from [Gopalan et al., 2011], to obtain our structural results, and use some non-trivial results from algebraic number theory to get the lower bound. We construct a family of at most s-sparse Boolean functions over ℤ_pⁿ, where p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is o(1/s). The "Granularity" result of Gopalan et al. implies that the absolute values of non-zero Fourier coefficients of any s-sparse Boolean valued function over ℤ₂ⁿ are Ω(1/s). So, our result shows that one cannot expect such a lower bound for general Abelian groups. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean function, which tests whether the given function is s-sparse, or ε-far from any sparse Boolean function, and it requires poly((ms)^φ(m),1/ε)-many queries. Further, we generalize the notion of degree of a Boolean function over an Abelian group 𝒢. We use it to prove an Ω(√s) lower bound on the query complexity of any adaptive sparsity testing algorithm.

Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits. Debordering is at the heart of understanding the difference between Valiant's determinant vs permanent conjecture, and Mulmuley and Sohoni's variation which uses border determinantal complexity. The debordering of matrix multiplication tensors by Bini played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Currently, very few debordering results are known. In this work, we study the question of debordering the border Waring rank of polynomials. Waring and border Waring rank are very well studied measures in the context of invariant theory, algebraic geometry, and matrix multiplication algorithms. For the first time, we obtain a Waring rank upper bound that is exponential in the border Waring rank and only linear in the degree. All previous known results were exponential in the degree. For polynomials with constant border Waring rank, our results imply an upper bound on the Waring rank linear in degree, which previously was only known for polynomials with border Waring rank at most 5.

Abstract For a polynomial f, a weighted sum-of-squares representation (SOS) has the form $$f = \sum_{i\in [s]} c_i f_i^2$$ f = ∑ i ∈ [ s ] c i f i 2 , where the weights $$c_i$$ c i are field elements. The size of the representation is the number of monomials that appear across the $$f_i$$ f i 's. Its minimum across all such decompositions is called the support-sum S(f) of f. For a univariate polynomial f of degree d of full support, a lower bound for the support-sum is $$S(f) \ge \sqrt d$$ S ( f ) ≥ d . We show that the existence of an explicit univariate polynomial f with support-sum just slightly larger than the lower bound, that is, $$S(f) \ge d^{0.5+\varepsilon}$$ S ( f ) ≥ d 0.5 + ε , for some $$\varepsilon > 0$$ ε > 0 , implies that $$\ne$$ ≠ , the major open problem in algebraic complexity. In fact, our proof works for some subconstant functions $$\varepsilon(d) > 0$$ ε ( d ) > 0 as well. We also consider the sum-of-cubes representation (SOC) of polynomials. We show that an explicit hard polynomial implies both blackbox-PIT is in , and $$\neq$$ ≠ .

Mulmuley and Sohoni (2001) proposed an ambitious program, the Geometric Complexity Theory (GCT), to prove $P\neq NP$ and related conjectures using algebraic geometry and representation theory. Gradually, GCT has introduced new structures and questions in complexity. GCT tries to capture the algebraic/geometric notion of ’approximation’ by defining border classes. Surprisingly, (Kumar ToCT’20) proved the universal power of the border of top-fanin- 2 depth-3 circuits $(\overline{\Sigma^{[2]}\Pi\Sigma})$; which is in complete contrast to its classical model. Recently, (Dutta,Dwivedi,Saxena, FOCS’21) put an upper bound, by showing that bounded-top-fanin border depth-3 circuits $(\overline{\Sigma^{[k]}\Pi\Sigma}$ for constant $k)$ can be computed by a polynomial-size algebraic branching program (ABP). It was left open to show an exponential separation between the class of ABPs and $\overline{\Sigma^{[k]}\Pi\Sigma}$. In this article, we show a strongly-exponential separation between any two consecutive border classes, $\overline{\Sigma^{[k]}\Pi\Sigma}$ and $\Sigma^{[k+1]}\Pi\Sigma$, establishing an optimal hierarchy of constant topfanin border depth- 3 circuits. Put in GCT language: we prove an exponential-hierarchy for padded- k-th-secant-varieties of the Chow variety of $\mathbb{F}^{n+1} $. This positively answers [Open question 2 of Dutta,Dwivedi,Saxena FOCS’21] and [Problem 8.10 with constant r, of Landsberg, Annal.Ferrara’15]. Full version: https://www.cse.iitk.ac.in/users/nitin/papers/exphierarchy.pdf

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of rootsris small but the multiplicities are exponentially large. Our method sets up a linear system inrunknowns and iteratively builds the roots as formal power series. For an algebraic circuit \( f(x_1,\ldots ,x_n) \) of sizes, we prove that each factor has size at most a polynomial insand the degree of the squarefree part off. Consequently, if \( f_1 \) is a \( 2^{\Omega (n)} \) -hard polynomial, then any nonzero multiple \( \prod _{i} f_i^{e_i} \) is equally hard for arbitrary positive \( e_i \) ’s, assuming that \( \sum _i\deg (f_i) \) is at most \( 2^{O(n)} \) . It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomialfof degree \( n^{O(1)} \) and formula (respectively, algebraic branching program) size \( n^{O(\log n)} \) , we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly( \( n^{\log n} \) ) time. Consequently, if the determinant requires an \( n^{\Omega (\log n)} \) size formula, then the same can be said about any of its nonzero multiples. In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation \( \tau \) , the polynomial \( f(\tau \overline{x}) \) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., [ 17 , 51 , 54 , 111 ]).

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (Σ^[k] Π Σ ∧) and sum-product-of-constant-degree-polynomials (Σ^[k] Π Σ Π^[δ]), for constants k, δ, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC'05; Kayal & Saxena, CCC'06; Saxena & Seshadhri, FOCS'10, STOC'11); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP'11; Saha,Saxena & Saptharishi, Comput.Compl.'13; Forbes, FOCS'15; Kumar & Saraf, CCC'16); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS'09; Shpilka, STOC'19; Peleg & Shpilka, CCC'20, STOC'21). We solve two of the basic underlying open problems in this work. We give the first polynomial-time PIT for Σ^[k] Π Σ ∧. Further, we give the first quasipolynomial time blackbox PIT for both Σ^[k] Π Σ ∧ and Σ^[k] Π Σ Π^[δ]. No subexponential time algorithm was known prior to this work (even if k = δ = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧.

Given polynomials f,g,h ∈ 𝔽[x₁,…,x_n] such that f = g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s,deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen'87 & WACT'16). The result is an exponential improvement over Strassen’s classic result (Strassen'73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)). Further, we show that any univariate polynomial family (f_d)_d, defined by the initial segment of the power series expansion of rational function g_d(x)/h_d(x) up to degree d (i.e. f_d = g_d/h_d od x^{d+1}), where circuit size of g is s_d and degree of g_d is at most d, can be computed by a circuit of size poly(s_d,deg(h_d),log d). We also show a hardness result when the degrees of the rational functions are high (i.e. Ω (d)), assuming hardness of the integer factorization problem. Finally, we extend this conditional hardness to simple algebraic functions as well, and show that for every prime p, there is an integral algebraic power series with its minimal polynomial satisfying a degree p polynomial equation, such that its initial segment is hard to compute unless integer factoring is easy, or a multiple of n! is easy to compute. Both, integer factoring and computation of multiple of n!, are believed to be notoriously hard. In contrast, we show examples of transcendental power series whose initial segments are easy to compute.

Border complexity of polynomials plays an integral role in GCT (Geometric Complexity Theory) approach to P versus NP. It tries to formalize the notion of ‘approximating a polynomial’ via limits (Bürgisser FOCS'01). This raises the open question whether border of VP is same as VP or not; as the approximation involves exponential precision, which may not be efficiently simulable. Recently (Kumar ToCT'20) proved the universal power of the border of top-fanin-2 depth-3 circuits. Here we answer some of the related open questions. We show that the border of bounded top-fanin-k depth-3 circuits, for constant k, is relatively easy- it can be computed by a polynomial size algebraic branching program (ABP). There were hardly any de-bordering results known for prominent models before our result. Moreover, we give the first quasipolynomial-time black-box identity test for the same. Prior best was in PSPACE (Forbes,Shpilka STOC'18). Also, with more technical work, we extend our results to depth-4. Our de-bordering paradigm is a multi-step process; in short we call it DiDIL -divide, derive, induct, with limit. It ‘almost’ reduces border top-fanin-k depth-3 circuits to special cases of read-once oblivious algebraic branching programs (ROABPs) in any-order. Full version: https://www.cse.iitk.ac.in/users/nitin/papers/border-depth3.pdf

For a polynomial f, we study the sum of squares representation (SOS), i.e. f = ∑_{i ∈ [s]} c_i f_i² , where c_i are field elements and the f_i’s are polynomials. The size of the representation is the number of monomials that appear across the f_i’s. Its minimum is the support-sum S(f) of f. For simplicity of exposition, we consider univariate f. A trivial lower bound for the support-sum of, a full-support univariate polynomial, f of degree d is S(f) ≥ d^{0.5}. We show that the existence of an explicit polynomial f with support-sum just slightly larger than the trivial bound, that is, S(f) ≥ d^{0.5+ε(d)}, for a sub-constant function ε(d) > ω(√{log log d/log d}), implies that VP ≠ VNP. The latter is a major open problem in algebraic complexity. A further consequence is that blackbox-PIT is in SUBEXP. Note that a random polynomial fulfills the condition, as there we have S(f) = Θ(d). We also consider the sum-of-cubes representation (SOC) of polynomials. In a similar way, we show that here, an explicit hard polynomial even implies that blackbox-PIT is in P.

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots r is small but the multiplicities are exponentially large. Our method sets up a linear system in r unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,…,xn) of size s we prove that each factor has size at most a polynomial in: s and the degree of the squarefree part of f. Consequently, if f1 is a 2Ω(n)-hard polynomial then any nonzero multiple ∏i fiei is equally hard for arbitrary positive ei’s, assuming that ∑ideg(fi) is at most 2O(n).