Is algebraic geometry constructive?

If you forget about all the layers of abstraction, algebraic geometry is, ultimately (and very roughly speaking), the study of polynomial equations in several variables, and of the geometric objects they define. So in a certain sense, whenever you're doing anything with multivariate polynomials, there's probably some algebraic geometry behind it; and conversely, algebraic geometry questions can generally be reduced, at least in principle, to "does this system of polynomial equations have a solution?". Now to go a (little) bit further, one should consider the field upon which these equations are defined, and more importantly, in which the solutions are sought. The whole point of algebraic geometry is that most of the formalism can be made uniform in the base field, or indeed, ring. But there are at least two main flavors:

• Geometric questions are over algebraically closed fields (and all that really matters, then, is the characteristic of the field). These questions are, in principle, algorithmically decidable (although the complexity can be very bad), at least if we bound every degree involved in the problem; Gröbner bases are a the key tool to solve these geometric problems in practice.

• Arithmetic questions are over any other field, typically the rational numbers (→ diophantine equations); for example, an arithmetic question could be "does the curve $x^n + y^n = z^n$ (where $x,y,z$ are homogeneous coordinates) have rational solutions beyond the obvious ones?". Arithmetic questions can be undecidable, so there is no universal tool like Gröbner bases to solve them. There is a subtle interplay between geometry and arithmetic (for example, in the simplest nontrivial case, that of curves, the fundamental geometric invariant, the genus $g$, determines very different behaviors on its rational points according as $g=0$, $g=1$ or $g\geq 2$).

Then there are some fields which are "not too far" from being algebraically closed, like the reals, the finite fields, and the $p$-adics. Here, it is still decidable in principle whether a system of polynomial equations has a solution, but the complexity is even worse than for algebraically closed fields (for finite fields, there is the obvious algorithm consisting of trying possible value). Some theory can help bring it down to a manageable level.

As for applications outside mathematics, they mostly fall in this "not too far from algebraically closed" region:

• Algebraic geometry over the reals has applications in robotics, algebraic statistics (which is part of mathematics but itself has applications to a wide variety of sciences), and computer graphics, for example.

• Algebraic geometry over finite fields has applications in cryptography (and perhaps more generally boolean circuits) and the construction of error-correcting codes.

But I would like to emphasize that the notion of "applications" is not quite clear-cut. Part of classical algebraic geometry is the theory of elimination (i.e., essentially given a system of polynomial equations in $n+k$ variables, find the equations in the $n$ first variables defining whether there exists a solution in the $k$ last): this is a very useful computational tool in a huge range of situations where polynomials or polynomial equations play any kind of rôle. For example, a number of years ago, I did some basic computations on the Kerr metric in general relativity (ultimately to produce such videos as this one): the computations themselves were differential-geometric in nature (and not at all sophisticated), but by remembering that, in the right coordinate system, everything is an algebraic function, and by using some elimination theory, I was able to considerably simplify some symbolic manipulations in those computations. I wouldn't call it an application of algebraic geometry to physics, but knowing algebraic geometry definitely help me not make a mess of the computations.

Yes it is! We have Gröbner basis algorithms that can answer the question of ideal membership and can be used to answer many other geometric questions. If you are interested in this further, Cox, Little O'Shea have a good introduction to algebraic geometry from this perspective.

Perhaps you will not consider this as a real-world application, but in recent years Theoretical Computer Science is using more and more Algebraic Geometry. For example, one main approach for attacking the "P vs NP" problem is based on Algebraic Geometry (see this Wikipedia page). There are various algorithms that rely on tools from algebraic geometry (see for example this paper). Other uses of Algebraic Geometry pop up in Cryptography, Coding Theory, and other sub-fields.