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Typical phrases used by academics in published articles on mathematics

The procedure is computationally expensive ....

The procedure is computationally difficult ....

. leads to .

We assume f is not identically zero.

In this paper we present ...

In Section 4 we ...

We end this paper with some directions for future research.

Background material and related work.

We say more about this below.


. finesses away the difficulties of working with actual points.

There is a classical notion rooted in the fields.

Theorems about results holding for these Zariski open sets .

Then let us critique this heuristic calculation.

We get the upper bound ..

The consideration of solving .. has been ostensibly neglected by the mainstream analysis so far.

It equally well goes for ..

In this sense, they also are an effective ..

Resultants that exploit sparsely are described in [21].

While this experience is anecdotal and ...

Its space complexity is surveyed in [41]

What the "appropriate vector prototype" amounts to needs additional specification.

We refer to [23] a continuation method to deal with solution manifolds.

Exploiting generic flatness in this way is the underlying approach of the work of .

Thus our algorithm achieves the same numerical goal as .

As one piece of evidence of its optimality, note that as a consequence we obtain the classical upper bound.

Then given general real or complex numbers xyz, all solutions on .

Thus it follows from Struda [14, Proposition on p. 24] that phi is flat.

Now X (U x V) is a local complete intersection, and since the map phi is proper with finite fibers, we conclude by Lemma 4, that this map is flat.

In the last column we list cpu times.

The cost would be of the same magnitude as on the d-row.

We must note that there exits an isotropic formulation of this problem.

This system was popularized in [16], and is by far the most notorious benchmark problem in polynomial system solving.

For a fixed e we provide lower and upper bounds on comp (e) of the form e1 where the exponent t is small for small variances d.

The complexity of this problem is of order e 1.

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