Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cs/0508113

Asymptotically fast polynomial matrix algorithms for multivariable systems

Claude-Pierre Jeannerod·Gilles Villard
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication.

Explore related subjects

cs.SCcs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Claude-Pierre Jeannerod, Gilles Villard. 2005-08-25. Asymptotically fast polynomial matrix algorithms for multivariable systems. https://arxiv.org/abs/cs/0508113

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On computing the determinant over a ring

An algorithm is presented for computing the determinant and the characteristic polynomial of a matrix $A$ over an arbitrary commutative ring R. The algorithm uses $O(n^ω)$ ring additions, subtractions and multiplications, where $ω>2$ is a feasible exponent for matrix multiplication, that is, such that two $n\times n$ matrices over R can be multiplied using $O(n^ω)$ ring operations. The algorithm involves no division and no zero test, and its control depends only on $n$: for each $n$ it is a straight-line program. The approach combines a polynomial matrix point of view and a block Krylov point of view on the characteristic polynomial, which makes Strassen's elimination of divisions efficient. A bivariate matrix $λI-L(t)$, obtained from $A$ by a homotopy, is compressed in $\log n$ steps, each of which halves the dimension while only doubling the degree in $λ$. The cost of the homotopy is controlled by truncating the power series in $t$ at an order that only grows with the compression steps, from a constant to $n$.

cs.SC↗
arXiv2026

Exact certification for conic optimization over quadratic modules: partitions of unity and recession directions

We consider linear conic problems whose cone is a product of truncated quadratic modules, the sum-of-squares (SOS) side of the moment-SOS hierarchy. A floating-point solution is turned into an exact rational certificate in three steps: rounding, repair of each block residual by a constant, and absorption of the constants along a recession direction of the problem. The repair relies on strict partitions of unity in the quadratic modules. We characterize their existence, relate the canonical one to the analytic center of the dual pseudo-moment cone, Christoffel-Darboux polynomials and Pell equations, and we derive closed-form rational partitions for balls and boxes. The loss incurred by the certification is the optimal value of a recession problem, equal to a support function of the masses of the dual feasible pseudomoments. The method is illustrated on polynomial optimization, volume computation and positivity of polynomials along D-finite trajectories.

cs.SC↗
arXiv2026

Where models of minimal height are hard to find: cubic rational maps with a multiple fixed point

Given a rational point of the moduli space of cubic rational maps, we seek a map in its class with the smallest integer coefficients. A recent algorithm (Shaska, arXiv:2610.33990) combines an exact descent and minimisation of the resultant with a reduction and a local search; it is nearly exact where the minimal height is known. Beyond that range, a certified deeper search shows that the instances in which a smaller model lies beyond its reach are not random: a classifier predicts them from the six invariants of the point, and asking what it relies on leads to an exact algebraic surface, the maps with a triple fixed point. Invariant theory and the stability hypothesis of reduction theory (Hutz and Stoll, 2019) then identify the larger locus that contains it: the hypersurface $ξ_1^3=6ξ_2^2$ of maps with a multiple fixed point, on which the fixed-point quartic is not stable. It holds most of the hard instances. On it the two invariants of the fixed-point quartic are $(ξ_1,ξ_2)=(6m^2,6m^3)$ for an integer $m\ge0$; we parametrise the integer points of the surface, and show that on its generic part the sign of $ξ_5$ is the sign of the cubic coefficient at the parabolic fixed point. A covariant defined on the locus, added to the reduction, resolves only a minority of the hard instances: they appear to be mainly a matter of the search's reach. These findings lead to a certified structured search that never returns a larger model than the original algorithm and finds most of the smaller models within reach, also at heights not used to design it, at a small multiple of its running time.

cs.SC↗