Undergraduate Texts in Mathematics are generally aimed at third-and fourthyear
undergraduate mathematics students at North American universities. These texts strive to …

In Chapter 1, we have seen how the algebra of the polynomial rings k [x1,..., xn] and the
geometry of affine algebraic varieties are linked. In this chapter, we will study the method of …

The world is continuous, but the mind is discrete. David Mumford We seek to bridge some
critical gaps between various? elds of mathematics by studying the interplay between the …

In this paper, we consider the counting function\({{\,\mathrm {{{\,\mathrm {\mathcal {E}}\,}} _
{{{\,\mathrm {\mathcal {P}}\,}}}}\,}}(y)=|{{\,\mathrm {\mathcal {P}}\,}} _ {y}\cap {{\,\mathrm …

Let M be a rational homology sphere plumbed 3–manifold associated with a connected
negative-definite plumbing graph. We show that its Seiberg–Witten invariants equal certain …

Let a polytope $ P $ be defined by a system $ A x\leq b $. We consider the problem of
counting the number of integer points inside $ P $, assuming that $ P $ is $\Delta $-modular …

We present a new tool to compute the number \bfA(b) of integer solutions to the linear
system x ≧ 0, A x= b, where the coefficients of A and b are integral. \bfA(b) is often described …

One of the main questions in the theory of normal surface singularities is to understand the
relations between their geometry and topology. The lattice cohomology is an important tool …

Given a real closed polytope $ P $, we first describe the Fourier transform of its indicator
function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform …

The defining feature of IP+ is the non-negativity of the input data A, b and c. While a variable
without sign restriction can be modeled with the help of a difference x= xp− xn of two non …