Symmetric polynomials are polynomials that are invariant under the action of the
symmetric group, and they play an integral role in mathematics. The space of quasiin-
variant polynomials, polynomials that are invariant under the action of the symmetric
group to a certain order, were introduced by Feigin and Veselov. These spaces are
modules over the ring of symmetric polynomials, and their Hilbert series in elds of
characteristic 0 were also computed by Feigin and Veselov.
In this paper, we study the Hilbert series of these spaces in elds of positive char-
acteristic. Braverman, Etingof, and Finkelberg recently introduced spaces of twisted
quasiinvariant polynomials, a generalization of quasiinvariant polynomials in which the
space is twisted by a monomial. We extend some of their results to spaces twisted by
a product of smooth functions and compute the Hilbert series of the space in certain
Because of its importance in number theory and singularity theory, the problem of nding
a polynomial sharp upper estimate of the number of positive integral points in an n-
dimensional (n 3) polyhedron has received attention by a lot of mathematicians. S. S.-T.
Yau proposed the upper estimate, so-called the Yau Number Theoretic Conjecture. The previous
results on the Yau Number Theoretic Conjecture in low dimension cases (n 6) have
been proved by using the sharp GLY conjecture. Unfortunately, it is only valid in low dimension.
The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new
method in . In this paper, the similar method has been applied to prove the Yau Number
Theoretic Conjecture for n = 8, but with more meticulous analyses. The main method of
proof is summing existing sharp upper bounds for the number of points in 7-dimensional
simplexes over the cross sections of eight-dimensional simplex. This reasearch project paves
the way for the proof of a fully general sharp upper bound for the number of lattice points
in a simplex. It also moves the mathematical community one step closer towards proving the
Yau Number Theoretic Conjecture in full generality. As an application, we give a sharper
estimate of the Dickman-De Bruijn function (x; y) for 5 y < 23, compared with the
result obtained by Ennola.
Polyhedral combinatorics has been a topic of interest in modern day‚Äôs computational geometry.
The founding of Steinitz‚Äôs Theorem in 1922 revealed consequential relations between graph
theory and polyhedral combinatorics. It allows us to better investigate on the topology of
convex polyhedrons. In this paper, we proposed an algorithm that generates a unique sequence
of points, using the vertices of a triangulated polyhedron, pre-determined by the selection of
the starting 3 vertices in the sequence. Following that, we discover an interesting relation
between the sequence and the volume of the polyhedron itself, in which we presented in the
form of a sufficient condition. To further investigate which polyhedrons generate sequences
that satisfy the sufficient condition, we study the problem in the context of graph theory, that
is, the explorer walk (corresponding to the sequence of vertices) in maximal planar graphs
(skeletons of triangulated convex polyhedrons). With that, we uncovered a family of maximal
planar graphs, called the explorer graphs, which exhibits volumetric properties in the
polyhedrons constructed from them, in regard to the explorer walk. In this paper, we also
introduce generalized methods of constructing explorer graphs of higher order from explorer
graphs of lower order, demonstrating the prevalence of explorer graphs. As the edges of a
maximal planar graph is of great importance in tracing an explorer walk, we investigate on the
line graph of maximal planar graphs, and re-establish a better definition of explorer graphs.
Lastly, our paper covers the edge contraction of explorer graphs, which allows us to solve the
volume of polyhedrons constructed from non-explorer graphs. For this, we presented a possible
bound for the minimum number of edge contractions a non-explorer graph requires from an
explorer graph. This will generalize the proposed method of finding volumes to any
triangulated convex polyhedron.