Let f (x) = 1+ ∑n=1 anx n be a formal power series with complex coefficients. Let {rn}n=1 be a sequence of nonzero integers. The Integer Power Product Expansion of f (x), denoted ZPPE, is ∏k=1(1+wkx k)rk . Integer Power Product Expansions enumerate partitions of multi-sets. The coefficients {wk}k=1 themselves possess interesting algebraic structure. This algebraic structure provides a lower bou...

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership P...

As noted by Green et al. several provenance analyses can be considered a special case of the general problem of computing formal polynomials resp. power-series as solutions of an algebraic system. Specific provenance is then obtained by means of evaluating the formal polynomial under a suitable homomorphism. Recently, we presented the idea of approximating the least solution of such algebraic s...

We construct an explicit filtration of the ring algebraic power series by constructible sets, measuring complexity these series. As example use this, we give a bound on dimension set bounded lying variety defined over field

Trees are a very basic object in computer science. They intervene in nearly a;ly domain, and they are studied for their own, or used to represent conveniently a given situation. There are at least three directions where investigations on trees themselves are motivated, and this for different reasons. First, the notion of tree is the basis of algebraic semantics (Nivat [19], Rosen [2*2], etc.). ...

If f = EW:-, a,,t” is a formal Laurent series with certain restrictions on the a,, then f = f-fof+ , where fcontains only negative powers of t, f+ contains only positive powers of t, and f. is independent of t. Applications include Lagrange’s formula for series reversion, the problem of counting lattice paths below a diagonal, and a theorem of Furstenberg that the diagonal of a rational power s...