Peter Scheiblechner

Lucerne University of Applied Sciences and Arts | HSLU · School of Engineering and Architecture

We prove a new lower bound for the decision complexity of a complex algebraic set in terms of the sum of its (compactly supported) Betti numbers, which is for the first time better than logarithmic. We apply this result to subspace arrangements including some well studied problems such as the knapsack and element distinctness problems.

We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algori...

Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. We prove a single exponential bound on the degrees of these polynomials for varieties of arbitrary dimension. More precisely, we show that the p-th de Rham cohomology of a smoot...

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic p>0, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to a non-degeneracy condition...

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford reg...

Cable trees are used in industrial products to transmit energy and information between different product parts. To this date, they are mostly assembled by humans and only few automated manufacturing solutions exist using complex robotic machines. For these machines, the wiring plan has to be translated into a wiring sequence of cable plugging opera...

Cable trees are used in industrial products to transmit energy and information between different product parts. To this date, they are mostly assembled by humans and only few automated manufacturing solutions exist using complex robotic machines. For these machines, the wiring plan has to be translated into a wiring sequence of cable plugging opera...

Let F be a holomorphic map whose components satisfy some polynomial relations. We present an algorithm for constructing Nash maps locally approximating F, whose components satisfy the same relations.

We prove an effective bound for the degrees of generators of the algebraic de Rham cohomology of smooth affine hypersurfaces. In particular, we show that the de Rham cohomology H_dR^p(X) of a smooth hypersurface X of degree d in C^n can be generated by differential forms of degree d^O(pn). This result is relevant for the algorithmic computation of...

We prove two versions of Stickelberger’s Theorem for positive dimensions and use them to compute the connected and irreducible components of a complex algebraic variety. If the variety is given by polynomials of degree ≤d in n variables, then our algorithms run in parallel (sequential) time (nlogd)^O(1) (d^{O(n^4)}). In the case of a hypersurface,...

We present an algorithm for counting the irreducible com- ponents of a complex algebraic variety defined by a fixed number of polynomials encoded as straight-line programs (slps). It runs in poly- nomial time in the Blum-Shub-Smale (BSS) model and in randomized parallel polylogarithmic time in the Turing model, both measured in the lengths and degr...

We give an efficient algorithm for counting the connected components of a complex affine hypersurface. Our algorithm runs in parallel time O(n2 log2 d) and sequential time dO(n). In the proof we use the effective Nullstellensatz for two polynomials, which we also prove by very elemen- tary methods.

We give a uniform method for the two problems #CCC and #ICC of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work eciently in parallel and can be implemented by algebraic circuits of polynomial depth, i.e., in parall...

We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147–191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex...