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On the pole placement problem for linear systems of arbitrary genus. (English) Zbl 0956.14023
V. G. Lomadze introduced [Acta Appl. Math. 34, No. 3, 305-312 (1994; Zbl 0798.93027)] linear systems associated to divisors and vector bundles on \(X\) of genus \(g>0\). We consider the pole placement problem in the setting and with the notation of this cited paper. Let \(F\) be a rank \(m\) effective vector bundle on an arbitrary complete, smooth, and irreducible algebraic curve \(X\) of genus \(g>0\); the pole placement problem (or PPP for short) asks if there is always a completely reachable linear system \(s\) with \(m\) inputs, \(F\) as associated Martin-Hermann vector bundle and the determinant \(\text{ch}(F)\) of \(F\) as characteristic divisor \(c(s)\).
In this paper for every integer \(m\geq 2\) and for every such curve \(X\) we will give a negative solution to PPP for the case of \(m\) inputs. Recall that for genus 0 PPP has always a solution. For the case of one input (i.e. the rank \(m=1\) case) V. G. Lomadze (loc. cit.; §7, theorem 3) has shown that PPP has always a solution. Indeed we will show that for \(g>0\) and \(m\geq 2\) the non solubility of PPP is rather the norm, not an exception. We will use heavily the notions introduced in the paper cited above.
14H60 Vector bundles on curves and their moduli
93B27 Geometric methods
14C20 Divisors, linear systems, invertible sheaves
93B55 Pole and zero placement problems
93C05 Linear systems in control theory
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