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Hironaka's original desingularization algorithm had constructive aspects but as I understand, also had aspects / computations which are not constructive and could not be done by hand or with a computer.

Currently, resolution

Resolution algorithms now typically are based upon blowing up the strata of some invariant (where heuristically the singularities are at their worst) and so they are constructive if you can compute the invariant. But these invariants might be for all intents and purposes completely incomputable, for example based on infinite amounts of data.

New algorithms have simplified many aspects on this, and indeed, algorithms have been implemented in computers, and it isn't so hard to write down all the various invariants as you follows modern algorithms.

You can see

http://mathoverflow.net/questions/4612/hironaka-desingularisation-theorem-new-proofs-in-literature

for more discussion.

Hironaka's original desingularization algorithm had constructive aspects but as I understand, also had aspects / computations which essentially are not constructive and could not be done by hand or with a computer.

These

Currently, resolution algorithms typically are based upon blowing up the strata of some invariant (where heuristically the singularities are at their worst) and so they are constructive if you can compute the invariant. But these invariants might be for all intents and purposes completely incomputable, for example based on infinite amounts of data.

New algorithms have simplified many aspects on this, and indeed, algorithms have been implemented in computers, and it isn't so hard to write down all the various invariants as you follows modern algorithms.

You can see

http://mathoverflow.net/questions/4612/hironaka-desingularisation-theorem-new-proofs-in-literature

for more discussion.

2 added 59 characters in body

Hironaka's original desingularization algorithm had constructive aspects but as I understand, also had aspects / computations which essentially could not be done by hand or with a computer.

These algorithms typically are based upon blowing up the strata of some invariant (where heuristically the singularities are at their worst) and so they are constructive if you can compute the invariant. But these invariants might be for all intents and purposes completely incomputable, for example based on infinite amounts of data.

New algorithms have simplified many aspects on this, and indeed, algorithms have been implemented in computers, and it isn't so hard to write down all the various invariants as you follows modern algorithms.

You can see

http://mathoverflow.net/questions/4612/hironaka-desingularisation-theorem-new-proofs-in-literature

for more discussion.