Document

**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP.
The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs.
Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.

Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2024.20, author = {Chatterjee, Prerona and Kush, Deepanshu and Saraf, Shubhangi and Shpilka, Amir}, title = {{Lower Bounds for Set-Multilinear Branching Programs}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.20}, URN = {urn:nbn:de:0030-drops-204167}, doi = {10.4230/LIPIcs.CCC.2024.20}, annote = {Keywords: Lower Bounds, Algebraic Branching Programs, Set-multilinear polynomials} }

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Extended Abstract

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters.
We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points.
Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank.
Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.

Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam. Discreteness of Asymptotic Tensor Ranks (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2024.20, author = {Bri\"{e}t, Jop and Christandl, Matthias and Leigh, Itai and Shpilka, Amir and Zuiddam, Jeroen}, title = {{Discreteness of Asymptotic Tensor Ranks}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.20}, URN = {urn:nbn:de:0030-drops-195483}, doi = {10.4230/LIPIcs.ITCS.2024.20}, annote = {Keywords: Tensors, Asymptotic rank, Subrank, Slice rank, Restriction, Degeneration, Diagonalization, SLOCC} }

Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given black-box access to a tensor of super-constant rank. Specifically, we obtain the following results:
1) A deterministic algorithm that reconstructs polynomials computed by Σ^{[k]}⋀^{[d]}Σ circuits in time poly(n,d,c) ⋅ poly(k)^{k^{k^{10}}},
2) A randomized algorithm that reconstructs polynomials computed by multilinear Σ^{[k]}∏^{[d]}Σ circuits in time poly(n,d,c) ⋅ k^{k^{k^{k^{O(k)}}}},
3) A randomized algorithm that reconstructs polynomials computed by set-multilinear Σ^{[k]}∏^{[d]}Σ circuits in time poly(n,d,c) ⋅ k^{k^{k^{k^{O(k)}}}},
where c = log q if 𝔽 = 𝔽_q is a finite field, and c equals the maximum bit complexity of any coefficient of f if 𝔽 is infinite.
Prior to our work, polynomial time algorithms for the case when the rank, k, is constant, were given by Bhargava, Saraf and Volkovich [Vishwas Bhargava et al., 2021].
Another contribution of this work is correcting an error from a paper of Karnin and Shpilka [Zohar Shay Karnin and Amir Shpilka, 2009] (with some loss in parameters) that also affected Theorem 1.6 of [Vishwas Bhargava et al., 2021]. Consequently, the results of [Zohar Shay Karnin and Amir Shpilka, 2009; Vishwas Bhargava et al., 2021] continue to hold, with a slightly worse setting of parameters. For fixing the error we systematically study the relation between syntactic and semantic notions of rank of Σ Π Σ circuits, and the corresponding partitions of such circuits.
We obtain our improved running time by introducing a technique for learning rank preserving coordinate-subspaces. Both [Zohar Shay Karnin and Amir Shpilka, 2009] and [Vishwas Bhargava et al., 2021] tried all choices of finding the "correct" coordinates, which, due to the size of the set, led to having a fast growing function of k at the exponent of n. We manage to find these spaces in time that is still growing fast with k, yet it is only a fixed polynomial in n.

Shir Peleg, Amir Shpilka, and Ben Lee Volk. Tensor Reconstruction Beyond Constant Rank. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 87:1-87:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{peleg_et_al:LIPIcs.ITCS.2024.87, author = {Peleg, Shir and Shpilka, Amir and Volk, Ben Lee}, title = {{Tensor Reconstruction Beyond Constant Rank}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {87:1--87:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.87}, URN = {urn:nbn:de:0030-drops-196157}, doi = {10.4230/LIPIcs.ITCS.2024.87}, annote = {Keywords: Algebraic circuits, reconstruction, tensor decomposition, tensor rank} }

Document

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

We say that two given polynomials f, g ∈ R[x_1, ..., x_n], over a ring R, are equivalent under shifts if there exists a vector (a_1, ..., a_n) ∈ Rⁿ such that f(x_1+a_1, ..., x_n+a_n) = g(x_1, ..., x_n). This is a special variant of the polynomial projection problem in Algebraic Complexity Theory.
Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SIAM J. Computing, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any t-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem.
Formally, for a ring R, let SparseShift_R be the following decision problem - Given a polynomial P(X), is there a vector 𝐚 such that P(X+𝐚) contains fewer monomials than P(X). We show that SparseShift_R is at least as hard as checking if a given system of polynomial equations over R[x_1,..., x_n] has a solution (Hilbert’s Nullstellensatz). As a consequence of this reduction, we get the following results.
1) SparseShift_ℤ is undecidable.
2) For any ring R (which is not a field) such that HN_R is NP_R-complete over the Blum-Shub-Smale model of computation, SparseShift_ is also NP_R-complete. In particular, SparseShift_ℤ is also NP_ℤ-complete.
We also study the gap version of the SparseShift_R and show the following.
1) For every function β:ℕ → ℝ_+ such that β ∈ o(1), N^β-gap-SparseShift_ℤ is also undecidable (where N is the input length).
2) For R = 𝔽_p, ℚ, ℝ or ℤ_q and for every β > 1 the β-gap-SparseShift_R problem is NP-hard. Furthermore, there exists a constant α > 1 such that for every d = O(1) in the sparse representation model, and for every d ≤ n^O(1) in the arithmetic circuit model, the α^d-gap-SparseShift_R problem is NP-hard when given polynomials of degree at most d, in O(nd) many variables, as input.

Suryajith Chillara, Coral Grichener, and Amir Shpilka. On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chillara_et_al:LIPIcs.STACS.2023.22, author = {Chillara, Suryajith and Grichener, Coral and Shpilka, Amir}, title = {{On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {22:1--22:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.22}, URN = {urn:nbn:de:0030-drops-176744}, doi = {10.4230/LIPIcs.STACS.2023.22}, annote = {Keywords: algebraic complexity, shift equivalence, polynomial equivalence, Hilbert’s Nullstellensatz, hardness of approximation} }

Document

**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if {𝒬} ⊂ ℂ[x₁.…,x_n] is a finite set, |{𝒬}| = m, of irreducible quadratic polynomials that satisfy the following condition
There is δ > 0 such that for every Q ∈ {𝒬} there are at least δ m polynomials P ∈ {𝒬} such that whenever Q and P vanish then so does a third polynomial in {𝒬}⧵{Q,P}.
then dim(span) = Poly(1/δ).
The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/δ) on the dimension (in the first work an upper bound of O(1/δ²) was given, which was improved to O(1/δ) in the second work).

Shir Peleg and Amir Shpilka. Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{peleg_et_al:LIPIcs.SoCG.2022.43, author = {Peleg, Shir and Shpilka, Amir}, title = {{Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.43}, URN = {urn:nbn:de:0030-drops-160515}, doi = {10.4230/LIPIcs.SoCG.2022.43}, annote = {Keywords: Sylvester-Gallai theorem, quadratic polynomials, Algebraic computation} }

Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩ = ∑_{j = 1}^r c_j |φ_j⟩ for c_j ∈ ℂ and stabilizer states φ_j. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states.
We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(√n) of Bravyi, Smith and Smolin [Bravyi et al., 2016]. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(√n/log n). This is the first non-trivial lower bound for approximate stabilizer rank.
Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of 𝔽₂ⁿ, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

Shir Peleg, Ben Lee Volk, and Amir Shpilka. Lower Bounds on Stabilizer Rank. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 110:1-110:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{peleg_et_al:LIPIcs.ITCS.2022.110, author = {Peleg, Shir and Volk, Ben Lee and Shpilka, Amir}, title = {{Lower Bounds on Stabilizer Rank}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {110:1--110:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.110}, URN = {urn:nbn:de:0030-drops-157063}, doi = {10.4230/LIPIcs.ITCS.2022.110}, annote = {Keywords: Quantum Computation, Lower Bounds, Stabilizer rank, Simulation of Quantum computers} }

Document

**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

In this paper we study polynomials in VP_{e} (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL^{aff}_n(𝔽) (the action of (A,b) ∈ GL^{aff}_n(𝔽) on a polynomial f ∈ 𝔽[x] is defined as (A,b)∘f = f(A^Tx+b)), are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. Specifically, we obtain the following results:
1) For C_n(ℓ_1(x),…,ℓ_n(x)) ≜ Trace(\begin{pmatrix} 𝓁₁(x) & 1 \\ 1 & 0 \end{pmatrix} ⋅ … ⋅ \begin{pmatrix} 𝓁_n(x) & 1 \\ 1 & 0 \end{pmatrix}), where the 𝓁_is are linearly independent linear functions, we construct a polynomial-sized interpolating set, and give a polynomial-time reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VP_e [Karl Bringmann et al., 2018], thus our construction gives the first polynomial-size interpolating set for a dense subclass of VP_e.
2) For polynomials of the form ANF_Δ(𝓁₁(x),…,𝓁_{4^Δ}(x)), where ANF_Δ(x) is the canonical read-once formula in alternating normal form, of depth 2Δ, and the 𝓁_is are linearly independent linear functions, we provide a quasipolynomial-size interpolating set. We also observe that the reconstruction algorithm of [Ankit Gupta et al., 2014] works for all polynomials in this class. This class is also dense in VP_e.
3) Similarly, we give a quasipolynomial-sized hitting set for read-once formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VP_e.
4) We give a quasipolynomial-sized hitting set for polynomials of the form f(𝓁₁(x),…,𝓁_{m}(x)), where f is an m-variate s-sparse polynomial. and the 𝓁_is are linearly independent linear functions in n ≥ m variables. This class is dense in ΣΠΣ.
5) For polynomials of the form ∑_{i=1}^{s}∏_{j=1}^{d}𝓁_{i,j}(x), where the 𝓁_{i,j}s are linearly independent linear functions, we construct a polynomial-sized interpolating set. We also observe that the reconstruction algorithm of [Neeraj Kayal and Chandan Saha, 2019] works for every polynomial in the class. This class is dense in ΣΠΣ. As VP = VNC², our results for VP_{e} translate immediately to VP with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made robust then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of k-independent polynomial maps) do not necessarily yield robust hitting sets.

Dori Medini and Amir Shpilka. Hitting Sets and Reconstruction for Dense Orbits in VP_{e} and ΣΠΣ Circuits. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 19:1-19:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{medini_et_al:LIPIcs.CCC.2021.19, author = {Medini, Dori and Shpilka, Amir}, title = {{Hitting Sets and Reconstruction for Dense Orbits in VP\underline\{e\} and \Sigma\Pi\Sigma Circuits}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {19:1--19:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.19}, URN = {urn:nbn:de:0030-drops-142930}, doi = {10.4230/LIPIcs.CCC.2021.19}, annote = {Keywords: Algebraic complexity, VP, VNP, algebraic circuits, algebraic formula} }

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Invited Talk

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Algebraic complexity is the field studying the intrinsic difficulty of algebraic problems in an algebraic model of computation, most notably arithmetic circuits. It is a very natural model of computation that attracted a large amount of research in the last few decades, partially due to its simplicity and elegance, but mostly because of its importance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, deciding whether P = BPP and more, will be easier to solve for arithmetic circuits.
In this talk I will give the basic definitions, explain the main questions and how they relate to their Boolean counterparts, and discuss what I view as promising approaches to tackling the most fundamental problems in the field.

Amir Shpilka. On Some Recent Advances in Algebraic Complexity (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 6:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{shpilka:LIPIcs.FSTTCS.2020.6, author = {Shpilka, Amir}, title = {{On Some Recent Advances in Algebraic Complexity}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {6:1--6:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.6}, URN = {urn:nbn:de:0030-drops-132472}, doi = {10.4230/LIPIcs.FSTTCS.2020.6}, annote = {Keywords: Algebraic Complexity, Arithmetic Circuits, Polynomial Identity Testing} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Σ^{[3]}ΠΣΠ^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials 𝒬 satisfy that for every two polynomials Q₁,Q₂ ∈ 𝒬 there is a subset 𝒦 ⊂ 𝒬, such that Q₁,Q₂ ∉ 𝒦 and whenever Q₁ and Q₂ vanish then ∏_{Q_i∈𝒦} Q_i vanishes, then the linear span of the polynomials in 𝒬 has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |𝒦| = 1.
An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics.

Shir Peleg and Amir Shpilka. A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 8:1-8:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{peleg_et_al:LIPIcs.CCC.2020.8, author = {Peleg, Shir and Shpilka, Amir}, title = {{A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {8:1--8:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.8}, URN = {urn:nbn:de:0030-drops-125606}, doi = {10.4230/LIPIcs.CCC.2020.8}, annote = {Keywords: Algebraic computation, Computational complexity, Computational geometry} }

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Complete Volume

**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

LIPIcs, Volume 137, CCC'19, Complete Volume

34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{shpilka:LIPIcs.CCC.2019, title = {{LIPIcs, Volume 137, CCC'19, Complete Volume}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019}, URN = {urn:nbn:de:0030-drops-108988}, doi = {10.4230/LIPIcs.CCC.2019}, annote = {Keywords: Theory of computation} }

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Front Matter

**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Front Matter, Table of Contents, Preface, Conference Organization

34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{shpilka:LIPIcs.CCC.2019.0, author = {Shpilka, Amir}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {0:i--0:xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.0}, URN = {urn:nbn:de:0030-drops-108221}, doi = {10.4230/LIPIcs.CCC.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP.

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk. Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{anderson_et_al:LIPIcs.CCC.2016.30, author = {Anderson, Matthew and Forbes, Michael A. and Saptharishi, Ramprasad and Shpilka, Amir and Volk, Ben Lee}, title = {{Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {30:1--30:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.30}, URN = {urn:nbn:de:0030-drops-58255}, doi = {10.4230/LIPIcs.CCC.2016.30}, annote = {Keywords: Algebraic Complexity, Lower Bounds, Derandomization, Polynomial Identity Testing} }

Document

**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are circuits from restricted boolean circuit classes. Essentially all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity).
Our main contributions are two general methods of converting certain algebraic lower bounds into proof complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean circuit lower bounds are turned into related proof-complexity ones, especially the "feasible interpolation" technique. We establish algebraic lower bounds of these forms for several explicit polynomials, against a variety of classes, and infer the relevant proof complexity bounds. These yield separations between IPS subsystems, which we complement by simulations to create a partial structure theory for IPS systems.
Our first method is a functional lower bound, a notion of Grigoriev and Razborov, which is a function f' from n-bit strings to a field, such that any polynomial f agreeing with f' on the boolean cube requires large algebraic circuit complexity. We develop functional lower bounds for a variety of circuit classes (sparse polynomials, depth-3 powering formulas, read-once algebraic branching programs and multilinear formulas) where f'(x) equals 1/p(x) for a constant-degree polynomial p depending on the relevant circuit class. We believe these lower bounds are of independent interest in algebraic complexity, and show that they also imply lower bounds for the size of the corresponding IPS refutations for proving that the relevant polynomial p is non-zero over the boolean cube. In particular, we show super-polynomial lower bounds for refuting variants of the subset-sum axioms in these IPS subsystems.
Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (non-zero) multiples require large algebraic circuit complexity. By extending known techniques, we give lower bounds for multiples for various restricted circuit classes such sparse polynomials, sums of powers of low-degree polynomials, and roABPs. These results are of independent interest, as we argue that lower bounds for multiples is the correct notion for instantiating the algebraic hardness versus randomness paradigm of Kabanets and Impagliazzo. Further, we show how such lower bounds for multiples extend to lower bounds for refutations in the corresponding IPS subsystem.

Michael A. Forbes, Amir Shpilka, Iddo Tzameret, and Avi Wigderson. Proof Complexity Lower Bounds from Algebraic Circuit Complexity. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{forbes_et_al:LIPIcs.CCC.2016.32, author = {Forbes, Michael A. and Shpilka, Amir and Tzameret, Iddo and Wigderson, Avi}, title = {{Proof Complexity Lower Bounds from Algebraic Circuit Complexity}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.32}, URN = {urn:nbn:de:0030-drops-58321}, doi = {10.4230/LIPIcs.CCC.2016.32}, annote = {Keywords: Proof Complexity, Algebraic Complexity, Nullstellensatz, Subset-Sum} }

Document

**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation
between black-box PIT and lower bounds we obtain lower bounds for these models.
For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial.
For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial.
A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a
hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas.
We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of
a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.
Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a
read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).

Rafael Oliveira, Amir Shpilka, and Ben Lee Volk. Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 304-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2015.304, author = {Oliveira, Rafael and Shpilka, Amir and Volk, Ben Lee}, title = {{Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {304--322}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.304}, URN = {urn:nbn:de:0030-drops-50548}, doi = {10.4230/LIPIcs.CCC.2015.304}, annote = {Keywords: Arithmetic Circuits, Derandomization, Polynomial Identity Testing} }

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