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This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. These algorithms are based on computation of the equations of multi-Rees algebras of ideals that generalises a result of Cox, Lin and Sosa. Using these equations we propose efficient algorithms for computation of mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity., Comment: The manuscript is prepared for the volume based on the International Conference in Honor of Prof. Ravi S. Kulkarni's 80th Birthday on Geometry, Groups and Mathematical Philosophy. arXiv admin note: text overlap with arXiv:1902.07384

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement., Comment: 16 pages

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal., Comment: 16 pages

Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and ${\mathcal R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $\mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function $HK(s)=\ell({\mathcal R}(\mathfrak n)/(\mathfrak n, \mathfrak n t)^{[s]})$ is given by a polynomial for all large $s.$ We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s).$, Comment: to appear in Commutative Algebra:150 years with Roger and Sylvia Wiegand, Contemporary Mathematics, Amer. Math. Soc

We prove that the Hilbert-Kunz function of the ideal $(I,It)$ of the Rees algebra $\mathcal{R}(I)$, where $I$ is an $\mathfrak{m}$-primary ideal of a $1$-dimensional local ring $(R,\mathfrak{m})$, is a quasi-polynomial in $e$, for large $e.$ For $s \in \mathbb{N}$, we calculate the Hilbert-Samuel function of the $R$-module $I^{[s]}$ and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal $(I,It)\mathcal{R}(I)$ when $I$ is a parameter ideal in a Cohen-Macaulay local ring of dimension $d \geq 2$, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case., Comment: Some changes in section 3, to appear in Communications in Algebra

We show unusual cooperative two-photon resonance between two-modes of field inside a photonic crystal cavity. The two-photon resonance occurs when two off resonant quantum dots emit one photon in each cavity mode and de-excite simultaneously. In the presence of phonon coupling the conditions for two-photon resonance change significantly. Using such two-photon two-mode interaction we propose to generate entangled state of two qutrits. The basis of a qutrit are formed by the state of the cavity mode containing $0$, $1$ and $2$ photons. We also discuss effect of phonon coupling on negativity of the generated entangled state., Comment: One of the coauthor does not want to publish on arxiv

We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity. The algorithms make use of the defining equations of the multi-Rees algebra of ideals, which are obtained by generalising a result of D. Cox. K.-N. Lin, and G. Sosa., Comment: Vivek Mukundan joined as a new co-author. The package has improved significantly from its earlier version

Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the normal reduction number., Comment: Theorem 5.5 corrected, to appear in Journal of Ramanujan Mathematical Society

In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained., Comment: 28 pages

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $\mathfrak m$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes., Comment: 24 pages, to appear in Mathematical Proceedings of the Cambridge Philosophical Society

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds., Comment: 19 pages, minor corrections done

We predict dominating cooperative two-photon emission from two quantum dots coupled with a single mode photonic crystal cavity. The cooperative two photon emission occurs when excitons in two off-resonantly coupled quantum dots decay simultaneously. The interaction with common cavity field leads to cavity induced two-photon emission which is strongly inhibited by electron phonon coupling. The interaction with common phonon bath produces phonon induced two-photon emission which increases on increasing temperature. For identical quantum dots cavity induced two-photon emission is negligible but phonon induced two-photon emission could be large., Comment: 6 pages

We discuss the generation of two types of entangled state of two photons-- noon state which is entangled in number and polarization, and polarization entangled state which is entangled in polarization and frequency. We consider a single quantum dot coupled with a bimodal cavity in strong coupling regime. We analyze the effect of exciton-phonon coupling on the concurrence of the generated entangled states. We find that for both states concurrence is maximum in the absence of the anisotropic energy gap between exciton states and remains unchanged in the presence of exciton-phonon coupling. However, for finite anisotropic energy gap concurrence decreases on the increasing temperature of phonon bath., Comment: 9 pages, 7 figures

We propose measurement based conditional generation of superposition of motional states of nanomechanical resonator. We consider a two level quantum mechanical system coupled with nanomechanical resonator through phonon exchange. An interaction, which produces shifts in the state of nanomechanical resonator depending on the state of the qubit, is realized by driving qubit through two resonant lasers. The measurement of the state of quantum mechanical system produces superposition states of nanomechanical resonator. We show that the quantum interference between states in superposition may lead to arbitrary large displacement in resonator. We also discuss decoherence of generated states using Wigner function., Comment: 5 pages 4 fig

In this expository paper we present proofs of Grothendieck-Serre Formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh's conjecture by Kummini and Masuti. We present an alternate proof of Huneke-Ooishi Theorem and a generalisation for multi-graded filtrations., Comment: 32 pages

We propose a scheme to realize a highly efficient solid state source of photon pairs using cavity-assisted stimulated Raman adiabatic passage (STIRAP) in a single quantum dot, where a single photon from pump pulse and two stokes photons from cavity mode drives the Raman transition. The Autler-Townes doublet, generated by using a resonant continuous wave laser between biexciton and exciton states, and two-photon-resonant transition through strongly coupled cavity mode are utilized to facilitate (1+2)type Raman transition in the quantum dot. We show in the case of weak pump although the probability of generating two photons in cavity mode is small without cavity damping but two-photon-resonant emission is enhanced by cavity damping within strong coupling regime. We also discuss spectrum of the generated photon pair and photon-photon correlations in the generated photon pair. The efficiency of two photon source could be more than 80\% in current experimental conditions.

We study relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal F= \{\mathcal F (\underline n)\}_{\underline n\in\mathbb Z^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalisation of the Kirby-Mehran Complex. An analysis of its homology leads to an analogue of Huneke's Fundamental Lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal F$ and reduction vectors with respect to complete reductions of $\mathcal F.$, Comment: 25 pages

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals., Comment: 16 pages