arXiv subjects

Christian Ikenmeyer

Publications and source records attributed to Christian Ikenmeyer.

At least 19 recordsLinked to original sources

arXiv2026

Counting in logarithmic space

We study the class $\#\mathsf{L}$ of functions counting accepting paths of non-deterministic log-space Turing machines and construct methods to prove containment in $\#\mathsf{L}$. We prove that a large number of classical combinatorial and number theoretic functions belong to this class: classical functions from enumerative combinatorics (multinomial coefficients, Catalan numbers, linear extensions of trees, Stirling numbers, etc), algebraic combinatorics (number of standard Young tableaux, etc), discrete geometry, number theoretic functions, representation theoretic multiplicities in a large class of cases. We show that $\mathrm{GL}_2$-plethysm coefficients of bounded length outer partition can be counted by log$^2$-space polytime verifiers. We pose numerous questions and conjectures on $\#\mathsf{L}$ containment and its generalizations, that suggest venues for conditionally disproving $\#\mathsf{P}$-completeness. While studying which combinatorial functions are in $\#\mathsf{P}$ provides a formal way of (dis)proving the existence of combinatorial interpretations, the lower class $\#\mathsf{L}$ serves as an analogue for functions computable in polynomial time.

math.CO↗

arXiv2026

Kronecker products and iterated matrix multiplication

We observe that the Kronecker product of tensors is the operation that converts the determinant polynomial into Cayley's first hyperdeterminant. We apply the Kronecker product to iterated matrix multiplication, which results in the hypercomputant, a VNP-complete and VW[1]-complete polynomial whose hardness we prove via the equivariance of the Kronecker product. The construction works over arbitrary commutative semirings and also for the tensor algebra and the exterior algebra. For the tensor algebra this gives a version of "noncommutative VNP", and for polynomials over the nonnegative real numbers this gives a version of "monotone VNP", each with the hypercomputant as the complete object. We take a parameterized complexity viewpoint and compare the noncommutative setting and the monotone setting. Using standard techniques we obtain optimal algebraic branching program width lower bounds in both settings, and these are notably not always the same. We also prove the polystability of the hypercomputant and that its isotypic components are characterized by their stabilizer.

cs.CC↗

arXiv2025

On the gradient of the coefficient of the characteristic polynomial

We prove the bivariate Cayley-Hamilton theorem, a powerful generalization of the classical Cayley-Hamilton theorem. The bivariate Cayley-Hamilton theorem has three direct corollaries that are usually proved independently: The classical Cayley-Hamilton theorem, the Girard-Newton identities, and the fact that the determinant and every coefficient of the characteristic polynomial has polynomially sized algebraic branching programs (ABPs) over arbitrary commutative rings. This last fact could so far only be obtained from separate constructions, and now we get it as a direct consequence of this much more general statement. The statement of the bivariate Cayley-Hamilton theorem involves the gradient of the coefficient of the characteristic polynomial, which is a generalization of the adjugate matrix. Analyzing this gradient, we obtain another new ABP for the determinant and every coefficient of the characteristic polynomial. This ABP has one third the size and half the width compared to the current record-holder ABP constructed by Mahajan-Vinay in 1997. This is the first improvement on this problem for 28 years. Our ABP is built around algebraic identities involving the first order partial derivatives of the coefficients of the characteristic polynomial, and does not use the ad-hoc combinatorial concept of clow sequences. This answers the 26-year-old open question by Mahajan-Vinay from 1999 about the necessity of clow sequences. We prove all results in a combinatorial way that on a first sight looks similar to Mahajan-Vinay, but it is closer to Straubing's and Zeilberger's constructions.

cs.CC↗

arXiv2025

Field-independent Kronecker-plethysm isomorphisms

We construct an explicit field-independent SL$_2$-equivariant isomorphism between an invariant space of tensors and a plethysm space. The existence of such an isomorphism was only known in characteristic 0, and only indirectly via character theory. Our isomorphism naturally extends the web of field-independent isomorphisms given by Hermite reciprocity, Hodge duality, and the Wronskian isomorphism. This is a characteristic free generalization of a classical situation in characteristic zero: certain rectangular Kronecker coefficients coincide with certain plethysm coefficients, and their non-negativiy proves the unimodality of the $q$-binomial coefficient. We also give a short combinatorial field-independent proof that the Hermite reciprocity map over the standard basis is a triangular matrix with 1s on the main diagonal.

math.RT↗

arXiv2025

Which graph motif parameters count?

For a fixed graph H, the function #IndSub(H,*) maps graphs G to the count of induced H-copies in G; this function obviously "counts something" in that it has a combinatorial interpretation. Linear combinations of such functions are called graph motif parameters and have recently received significant attention in counting complexity after a seminal paper by Curticapean, Dell and Marx (STOC'17). We show that, among linear combinations of functions #IndSub(H,*) involving only graphs H without isolated vertices, precisely those with positive integer coefficients maintain a combinatorial interpretation. It is important to note that graph motif parameters can be nonnegative for all inputs G, even when some coefficients are negative. Formally, we show that evaluating any graph motif parameter with a negative coefficient is impossible in an oracle variant of #P, where an implicit graph is accessed by oracle queries. Our proof follows the classification of the relativizing closure properties of #P by Hertrampf, Vollmer, and Wagner (SCT'95) and the framework developed by Ikenmeyer and Pak (STOC'22), but our application of the required Ramsey theorem turns out to be more subtle, as graphs do not have the required Ramsey property. Our techniques generalize from graphs to relational structures, including colored graphs. Vastly generalizing this, we introduce motif parameters over categories that count occurrences of sub-objects in the category. We then prove a general dichotomy theorem that characterizes which such parameters have a combinatorial interpretation. Using known results in Ramsey theory for categories, we obtain a dichotomy for motif parameters of finite vector spaces as well as parameter sets.

cs.CC↗

arXiv2025

Strassen's algorithm via orbit flip graphs

We give a short proof for Strassen's result that the rank of the 2 by 2 matrix multiplication tensor is at most 7. The proof requires no calculations and also no pattern matching or other type of nontrivial verification, and is based solely on properties of a specific order 6 group action. Our proof is based on the recent combination of flip graph algorithms and symmetries.

cs.DS↗

arXiv2025

Fractional Divisibility of Spheres with Partially Generic Sets of Rotations

We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that its translations by $g_1,...,g_r$ sum up to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r/2$ rotations are ``generic".

math.MG↗

arXiv2024

Algebraic metacomplexity and representation theory

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 Poincar\'e-Birkhoff-Witt theorem for Lie algebras and on Gelfand-Tsetlin theory, for which we give the necessary comprehensive background.

cs.CC↗

arXiv2024

Functional Closure Properties of Finite $\mathbb{N}$-weighted Automata

We determine all functional closure properties of finite $\mathbb{N}$-weighted automata, even all multivariate ones, and in particular all multivariate polynomials. We also determine all univariate closure properties in the promise setting, and all multivariate closure properties under certain assumptions on the promise, in particular we determine all multivariate closure properties where the output vector lies on a monotone algebraic graph variety.

cs.CC↗

arXiv2024

Fixed-parameter debordering of Waring rank

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.

cs.CC↗

arXiv2023

Homogeneous Algebraic Complexity Theory and Algebraic Formulas

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.

cs.CC↗

arXiv2023

A remark on the quantum complexity of the Kronecker coefficients

We prove that the computation of the Kronecker coefficients of the symmetric group is contained in the complexity class #BQP. This improves a recent result of Bravyi, Chowdhury, Gosset, Havlicek, and Zhu. We use only the quantum computing tools that are used in their paper and additional classical representation theoretic insights. We also prove the analogous result for the plethysm coefficients.

quant-ph↗

arXiv2023

All Kronecker coefficients are reduced Kronecker coefficients

We settle the question of where exactly the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of a question by Stanley from 2000 and a question by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is $NP$-hard, and computing them is $\#P$-hard under parsimonious many-one reductions.

math.CO↗

arXiv2022

Karchmer-Wigderson Games for Hazard-free Computation

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 significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.

cs.CC↗

arXiv2022

Geometric complexity theory for product-plus-power

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.

cs.CC↗

arXiv2022

Degree-restricted strength decompositions and algebraic branching programs

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.

cs.CC↗

arXiv2022

Positivity of the symmetric group characters is as hard as the polynomial time hierarchy

We prove that deciding the vanishing of the character of the symmetric group is $C_=P$-complete. We use this hardness result to prove that the the square of the character is not contained in $\#P$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is $PP$-complete under many-one reductions, and hence $PH$-hard under Turing-reductions.

math.CO↗

arXiv2022

What is in #P and what is not?

For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not. We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations. We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of algebraic identities and inequalities. We investigate #TFNP and obtain oracle separations that prove the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1.

cs.CC↗