Semialgebraic range searching, arguably the most general version of range searching, is a fundamental problem in computational geometry. In the problem, we are to preprocess a set of points in R^D such that the subset of points inside a semialgebraic region described by a constant number of polynomial inequalities of degree ? can be found efficiently. Relatively recently, several major advances were made on this problem. Using algebraic techniques, “near-linear space” data structures [6, 18] with almost optimal query time of Q(n) = O(n^1-1/D+o⁽¹⁾) were obtained. For “fast query” data structures (i.e., when Q(n) = n^o(1)), it was conjectured that a similar improvement is possible, i.e., it is possible to achieve space S(n) = O(n^D+o⁽¹⁾). The conjecture was refuted very recently by Afshani and Cheng [3]. In the plane, i.e., D = 2, they proved that S(n) = ?(n^?+1-o⁽¹⁾/Q(n)^(?+3)?/²) which shows ?(n^?+1-o⁽¹⁾) space is needed for Q(n) = n^o(1). While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of n or Q(n) seem to be tight even for D = 2, as the best known upper bounds have S(n) = O(n^m+o⁽¹⁾/Q(n)⁽m-1)D/^(D-1)) where m = (^D_D^+?) - 1 = ?(?^D) is the maximum number of parameters to define a monic degree-? D-variate polynomial, for any constant dimension D and degree ?. In this paper, we resolve two of the issues: we prove a lower bound in D-dimensions, for constant D, and show that when the query time is n^o(1) + O(k), the space usage is ?(n^m-o⁽¹⁾), which almost matches the Õ(n^m) upper bound and essentially closes the problem for the fast-query case, as far as the exponent of n is considered in the pointer machine model. When considering the exponent of Q(n), we show that the analysis in [3] is tight for D = 2, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or to obtain better lower bounds a new fundamentally different input set needs to be constructed.