Let's say I'm a physicalist. Usually then, I will think of abstract objects as mental constructs, which are dictated by mental states, which, as a physicalist will say, are manifestations of physical things (e.g. a bunch of elementary particles, which we call the brain, interacting with each other inside my head). Given this, can we say that abstract objects (e.g. numbers, etc) for a physicalist physically exist?
You can say that a neural representation of abstract objects physically exist; that wouldn't be controversial for a physicalist.
You need only postulate, however, that certain computational systems (like our brains) recognize similarities between distinct objects; this seems rather weaker than what one normally means by an object (abstract or otherwise) existing. For instance, one could theoretically remember every case of an item that was called a "chair" and perform a pairwise comparison between a new object that may be a chair and all of those; and if this new object is as similar to the other chairs as they are to each other, and there is no other label with a set of exemplars that is a better mtach to the object you've got, then you have a chair. This is a perfectly functional system for naming and recognizing chairs, but what in this description is this "abstract chair"?
So it is not necessary to suppose that abstract objects exist, even if we act as though there is such an abstract object. Ultimately, a physicalist probably ought to defer to evidence regarding whether abstract objects effectively exist in brains or not (the physical world certainly contains regularities, but one is under no obligation to create an abstraction to instantiate those regularities).
From Gottlob Frege, Basic Laws of Arithmetic (1893), new english translation by Philip Ebert & Marcus Rossberg, Oxford UP 2013, page xxiv :
We have to think of cognition as an activity that does not create what is cognised, but grasp what is already there.
When I grasp a pencil, many things take place in my body: stimulation of the nerves, changes in the tension and the pressure of muscles, tendons and bones, changes in the circulation of blood. The sum of these processes, however, is not the pencil, nor do they create it. The latter has being independently of these processes. It is essential to grasping that there is something which is grasped; the inner changes alone are not the grasping. Similarly, what we mentally apprehend has being independently of this activity, of the ideas and their changes that are part of or accompany the apprehension; it is neither the sum of these processes nor is it created as part of our mental life.
All we need is a neural representation of criteria to be satisfied. Since those criteria to be satisfied can find themselves to be multiply realizable, or multiply represented, in various nervous systems, or various minds, we as physicalists need not commit ourselves to the mind-independent existence, spatiotemporal or not, of what take the place of abstract objects in our ontic discourse.