By Peter Bürgisser

One of crucial and winning theories in computational complicated ity is that of NP-completeness. This discrete conception is predicated at the Turing desktop version and achieves a category of discrete computational prob lems based on their algorithmic hassle. Turing machines formalize al gorithms which function on finite strings of symbols over a finite alphabet. against this, in algebraic types of computation, the elemental computational step is an mathematics operation (or comparability) of components of a hard and fast box, for in stance of actual numbers. Hereby one assumes precise mathematics. In 1989, Blum, Shub, and Smale [12] mixed present algebraic types of computation with the idea that of uniformity and constructed a conception of NP-completeness over the reals (BSS-model). Their paper created a renewed curiosity within the box of algebraic complexity and initiated new study instructions. the last word objective of the BSS-model (and its destiny extensions) is to unite classical dis crete complexity concept with numerical research and therefore to supply a deeper beginning of clinical computation (cf. [11, 101]). Already ten years sooner than the BSS-paper, Valiant [107, one hundred ten] had proposed an analogue of the idea of NP-completeness in a completely algebraic body paintings, in reference to his well-known hardness consequence for the everlasting [108]. whereas the a part of his idea in response to the Turing procedure (#P-completeness) is now regular and recognized one of the theoretical desktop technology com munity, his algebraic completeness outcome for the permanents got less attention.

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**Example text**

Our construction is related to the one in Kirkpatrick and Hell [65]. g. assume that m ~ 3. Take an m-cycle and duplicate one of its nodes including its adjacent edges. Let us denote the resulting graph by Dm and call the duplicated nodes the inner and outer connector, respectively. Now take m disjoint copies of Dm and join their inner connectors to form an m-cycle. '" Cm (see Fig. 1). The graph 8 m has a unique covering C+ by m-cycles. If we remove all the connectors (including their adjacent edges), then the resulting graph also has a unique covering by m-cycles, called C _.

Spanning cycles) in a graph. This should not be confused with the directed Hamilton cycle polynomial HC n which enumerates the directed Hamilton cycles in a digraph. We leave it to the reader to check that UHC m is a projection of UHC n if n 2: m+ 2. By Thm. 10 we know already that the families (GF(Kn, VI)) and (HC n ) are complete. As a byproduct of our later investigations in Sect. 5, we will obtain that the family (UHC n ) of undirected Hamilton cycle polynomials is complete as well (see Cor.

In other words, this number equals the permanent perK[211] = O. We conclude that WI = O. In a similar way, using the above mentioned properties of K, we can show that W 2 = ... = W 4 = O. W5 turns out to be the sum of the weights of all cycle covers of G which neither contain c nor c', and W6 is the sum of the weights of all cycle covers of G which contain both c and c'. This proves the lemma. 0 In our proof of Prop. 17 we need further auxiliary digraphs: the rosettes R(J-L) defined for a positive integer J-L (cf.