<p><strong>Is there a way of forcing a (child) class to have constructors with particular signatures or particular static methods in C# or Java?</strong></p>

<p>You can't obviously use interfaces for this, and I know that it will have a limited usage. One instance in which I do find it useful is when you want to enforce some design guideline, for example:</p>

<p><strong>Exceptions</strong><br>
They should all have the four canonical constructors, but there is no way to enforce it. You have to rely on a tool like FxCop (C# case) to catch these.</p>

<p><strong>Operators</strong><br>
There is no contract that specifies that two classes can be summed (with operator+ in C#)</p>

<p>Is there any design pattern to work around this limitation?
What construct could be added to the <em>language</em> to overcome this limitation in future versions of C# or Java?</p>

Using generics you can force a type argument to have a parameterless constructor - but that's about the limit of it.
Other than in generics, it would be tricky to actually
use
these restrictions even if they existed, but it could sometimes be useful for type parameters/arguments. Allowing static members in interfaces (or possibly static interfaces) could likewise help with the "generic numeric operator" issue.
I
wrote about this
a little while ago when facing a similar problem.