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

We study algebraic complexity classes and their complete polynomials under 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 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.

Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov. Homogeneous Algebraic Complexity Theory and Algebraic Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 43:1-43:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dutta_et_al:LIPIcs.ITCS.2024.43, author = {Dutta, Pranjal and Gesmundo, Fulvio and Ikenmeyer, Christian and Jindal, Gorav and Lysikov, Vladimir}, title = {{Homogeneous Algebraic Complexity Theory and Algebraic Formulas}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {43:1--43:23}, 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.2024.43}, URN = {urn:nbn:de:0030-drops-195713}, doi = {10.4230/LIPIcs.ITCS.2024.43}, annote = {Keywords: Homogeneous polynomials, Waring rank, Arithmetic formulas, Border complexity, Geometric Complexity theory, Symmetric polynomials} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games.
Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.

Christian Ikenmeyer, Balagopal Komarath, and Nitin Saurabh. Karchmer-Wigderson Games for Hazard-Free Computation. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 74:1-74:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ikenmeyer_et_al:LIPIcs.ITCS.2023.74, author = {Ikenmeyer, Christian and Komarath, Balagopal and Saurabh, Nitin}, title = {{Karchmer-Wigderson Games for Hazard-Free Computation}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {74:1--74:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.74}, URN = {urn:nbn:de:0030-drops-175775}, doi = {10.4230/LIPIcs.ITCS.2023.74}, annote = {Keywords: Hazard-free computation, monotone computation, Karchmer-Wigderson games, communication complexity, lower bounds} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

We analyze Kumar’s recent quadratic algebraic branching program size lower bound proof method (CCC 2017) for the power sum polynomial. We present a refinement of this method that gives better bounds in some cases.
The lower bound relies on Noether-Lefschetz type conditions on the hypersurface defined by the homogeneous polynomial. In the explicit example that we provide, the lower bound is proved resorting to classical intersection theory.
Furthermore, we use similar methods to improve the known lower bound methods for slice rank of polynomials. We consider a sequence of polynomials that have been studied before by Shioda and show that for these polynomials the improved lower bound matches the known upper bound.

Fulvio Gesmundo, Purnata Ghosal, Christian Ikenmeyer, and Vladimir Lysikov. Degree-Restricted Strength Decompositions and Algebraic Branching Programs. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 20:1-20:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gesmundo_et_al:LIPIcs.FSTTCS.2022.20, author = {Gesmundo, Fulvio and Ghosal, Purnata and Ikenmeyer, Christian and Lysikov, Vladimir}, title = {{Degree-Restricted Strength Decompositions and Algebraic Branching Programs}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.20}, URN = {urn:nbn:de:0030-drops-174127}, doi = {10.4230/LIPIcs.FSTTCS.2022.20}, annote = {Keywords: Lower bounds, Slice rank, Strength of polynomials, Algebraic branching programs} }

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

Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant’s determinant vs permanent conjecture, but recently Bürgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-tableau is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank. As a structural side result we prove that border Waring rank is bounded from above by the ABP width complexity.

Markus Bläser, Julian Dörfler, and Christian Ikenmeyer. On the Complexity of Evaluating Highest Weight Vectors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 29:1-29:36, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blaser_et_al:LIPIcs.CCC.2021.29, author = {Bl\"{a}ser, Markus and D\"{o}rfler, Julian and Ikenmeyer, Christian}, title = {{On the Complexity of Evaluating Highest Weight Vectors}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {29:1--29:36}, 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.2021.29}, URN = {urn:nbn:de:0030-drops-143036}, doi = {10.4230/LIPIcs.CCC.2021.29}, annote = {Keywords: Algebraic complexity theory, geometric complexity theory, algebraic branching program, Waring rank, border Waring rank, representation theory, highest weight vector, treewidth} }

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

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(ℂ)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions).
We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.

Ankit Garg, Christian Ikenmeyer, Visu Makam, Rafael Oliveira, Michael Walter, and Avi Wigderson. Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 12:1-12:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{garg_et_al:LIPIcs.CCC.2020.12, author = {Garg, Ankit and Ikenmeyer, Christian and Makam, Visu and Oliveira, Rafael and Walter, Michael and Wigderson, Avi}, title = {{Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {12:1--12:17}, 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.2020.12}, URN = {urn:nbn:de:0030-drops-125645}, doi = {10.4230/LIPIcs.CCC.2020.12}, annote = {Keywords: generators for invariant rings, succinct encodings} }

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

Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most k is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width.
It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most k is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes.

Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 21:1-21:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blaser_et_al:LIPIcs.CCC.2020.21, author = {Bl\"{a}ser, Markus and Ikenmeyer, Christian and Mahajan, Meena and Pandey, Anurag and Saurabh, Nitin}, title = {{Algebraic Branching Programs, Border Complexity, and Tangent Spaces}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {21:1--21:24}, 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.2020.21}, URN = {urn:nbn:de:0030-drops-125733}, doi = {10.4230/LIPIcs.CCC.2020.21}, annote = {Keywords: Algebraic Branching Programs, Border Complexity, Tangent Spaces, Lower Bounds, Geometric Complexity Theory, Flows, VQP, VNP} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. We provide the first toy setting in which a separation can be achieved for a family of polynomials via these multiplicities.
Mulmuley and Sohoni’s papers also conjecture that the vanishing behavior of multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. We provide first finite settings where a separation via multiplicities can be achieved, while the separation via occurrences is provably impossible. These settings are surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety).
As a side result we prove a slight generalization of Hermite’s reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.

Julian Dörfler, Christian Ikenmeyer, and Greta Panova. On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 51:1-51:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dorfler_et_al:LIPIcs.ICALP.2019.51, author = {D\"{o}rfler, Julian and Ikenmeyer, Christian and Panova, Greta}, title = {{On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.51}, URN = {urn:nbn:de:0030-drops-106276}, doi = {10.4230/LIPIcs.ICALP.2019.51}, annote = {Keywords: Algebraic complexity theory, geometric complexity theory, Waring rank, plethysm coefficients, occurrence obstructions, multiplicity obstructions} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the class of polynomials that can be approximated arbitrarily closely by polynomials in VP_e. We describe VP_e-bar with a strikingly simple complete polynomial (in characteristic different from 2) whose recursive definition is similar to the Fibonacci numbers. Further understanding this polynomial seems to be a promising route to new formula lower bounds.
Our methods are rooted in the study of ABPs of small constant width. In 1992 Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP_e-bar.
As a natural continuation of this work we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.

Karl Bringmann, Christian Ikenmeyer, and Jeroen Zuiddam. On Algebraic Branching Programs of Small Width. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 20:1-20:31, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bringmann_et_al:LIPIcs.CCC.2017.20, author = {Bringmann, Karl and Ikenmeyer, Christian and Zuiddam, Jeroen}, title = {{On Algebraic Branching Programs of Small Width}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {20:1--20:31}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.20}, URN = {urn:nbn:de:0030-drops-75217}, doi = {10.4230/LIPIcs.CCC.2017.20}, annote = {Keywords: algebraic branching programs, algebraic complexity theory, border complexity, formula size, iterated matrix multiplication} }

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