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We find that the abundance of large clusters of nucleons in neutron-rich matter at sub-nuclear density is greatly reduced by finite temperature effects when matter is close to beta-equilibrium. Large nuclei and exotic non-spherical nuclear configurations called pasta, favored in the vicinity of the transition to uniform matter at $T=0$, dissolve at relatively low temperature. For matter close to beta-equilibrium we find that the pasta melting temperature is $T_m^\beta\simeq 4\pm 1$~MeV for realistic equations of state. The mechanism for pasta dissolution is discussed, and in general $T_m^\beta$ is shown to be sensitive to the proton fraction. We find that coherent neutrino scattering from nuclei and pasta makes a modest contribution to the opacity under the conditions encountered in supernovae and neutron star mergers. Implications for neutrino signals from galactic supernovae are briefly discussed., Comment: 9 pages, 7 figures, 1 table

A new method is proposed for determining the ground state wave function of a quantum many-body system on a quantum computer, without requiring an initial trial wave function that has good overlap with the true ground state. The technique of Spectral Combing involves entangling an arbitrary initial wave function with a set of auxiliary qubits governed by a time dependent Hamiltonian, resonantly transferring energy out of the initial state through a plethora of avoided level crossings into the auxiliary system. The number of avoided level crossings grows exponentially with the number of qubits required to represent the Hamiltonian, so that the efficiency of the algorithm does not rely on any particular energy gap being large. We give an explicit construction of the quantum gates required for the realization of this procedure and explore the results of classical simulations of the algorithm on a small quantum computer with up to 8 qubits. We show that for certain systems and comparable results, Spectral Combing requires fewer quantum gates to realize than the Quantum Adiabatic Algorithm.

We extend the prediction range of Pionless Effective Field Theory with an analysis of the ground state of $^{16}$O in leading order. To renormalize the theory, we use as input both experimental data and lattice QCD predictions of nuclear observables, which probe the sensitivity of nuclei to increased quark masses. The nuclear many-body Schr\"odinger equation is solved with the Auxiliary Field Diffusion Monte Carlo method. For the first time in a nuclear quantum Monte Carlo calculation, a linear optimization procedure, which allows us to devise an accurate trial wave function with a large number of variational parameters, is adopted. The method yields a binding energy of $^{4}$He which is in good agreement with experiment at physical pion mass and with lattice calculations at larger pion masses. At leading order we do not find any evidence of a $^{16}$O state which is stable against breakup into four $^4$He, although higher-order terms could bind $^{16}$O., Comment: 11 pages, 4 figures

Several moduli spaces parametrizing linear subspaces of the projective space are cut out by linear and quadratic equations in their natural embedding: Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper is to prove that a similar statement holds when one replaces linear subspaces with algebraic subschemes of the projective space. We exhibit equations of degree 1 and 2 that define schematically the Hilbert schemes $\mathbf{Hilb}^{p}_{\mathbb P^n}$ for all (possibly nonconstant) Hilbert polynomials $p$. The equations are reminiscent of the Pl\"ucker relations on the Grassmannians: they are built formally with permutations on indexes on the Pl\"ucker coordinates. Our method relies on a new construction of the Hilbert scheme as a quotient of a scheme of quiver representations., Comment: Simplified and enhanced version. We prove that the bound $R>=r$ for the validity of our equations is sharp using considerations on Castelnuvo-Mumford-Gotzmann regularity. We explain the meaning of these equations when $R

Let $\mathcal{H}$ be the Hermitian curve defined over a finite field $\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on $\mathcal{H}$ complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. \texttt{DegRevLex}. For most Hermitian codes, and especially for all those with distance $d\geq q^2-q$ studied in [1], one of the two curves is always the Hermitian curve $\mathcal{H}$ itself, while if $d

We calculate the thermal conductivity of electrons for the strongly correlated multi-component ion plasma expected in the outer layers of neutron star's crust employing a Path Integral Monte Carlo (PIMC) approach. This allows us to isolate the low energy response of the ions and use it to calculate the electron scattering rate and the electron thermal conductivity. We find that the scattering rate is enhanced by a factor 2-4 compared to earlier calculations based on the simpler electron-impurity scattering formalism. This findings directly impacts the interpretation of thermal relaxation observed in transiently accreting neutron stars and has implications for the composition and nuclear reactions in the crust that occur during accretion., Comment: Revised version addressing referee comments

Let $ \Bbbk$ be a field of arbitrary characteristic, $A$ a Noetherian $ \Bbbk$-algebra and consider the polynomial ring $A[\mathbf x]=A[x_0,\dots,x_n]$. We consider homogeneous submodules of $A[\mathbf x]^m$ having a special set of generators: a marked basis over a quasi-stable module. Such a marked basis inherits several good properties of a Gr\"obner basis, including a Noetherian reduction relation. The set of submodules of $A[\mathbf x]^m$ having a marked basis over a given quasi-stable module has an affine scheme structure that we are able to exhibit. Furthermore, the syzygies of a module generated by such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in $\oplus^{m'}_{i=1} A[\mathbf x](-d_i)$). We apply the construction of marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we can explicitely construct (up to the action of a general linear group) an open cover of the corresponding Quot functor made up of open functors represented by affine schemes. This gives a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points., Comment: 28 pages, exposition improved. This version contains the results of the previous one, and also the application to Quot schemes

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper., Comment: 36 pages. New title and substantial improvements to the presentation according to the comments of the reviewers

In this paper we present a geometrical characterization for the minimum-weight codewords of the Hermitian codes over the fields $\mathbb{F}_{q^2}$ in the third and fourth phase, namely with distance $d \geq q^2-q$. We consider the unique writing $\mu q + \lambda (q+1)$ of the distance $d$ with $\mu, \lambda$ non negative integers, and $\mu \leq q$, and prove that the minimum-weight codewords correspond to complete intersection divisors cut on the Hermitian curve $\mathcal{H}$ by curves $\mathcal X$ of degree $\mu+\lambda$ having $x^\mu y^\lambda$ as leading term w.r.t. the $\texttt{DegRevLex}$ term ordering (with $y>x$). Moreover, we show that any such curve $\mathcal X$ corresponds to minimum-weight codewords provided that the complete intersection divisor $\mathcal{H}\cap \mathcal X$ is made of simple $\mathbb{F}_{q^2}$-points. Finally, using this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command $\mathtt{MinimumWords}$.

We explore the use of mean field models to approximate microscopic nuclear equations of state derived from chiral effective field theory across the densities and temperatures relevant for simu- lating astrophysical phenomena such as core-collapse supernovae and binary neutron star mergers. We consider both relativistic mean field theory with scalar and vector meson exchange as well as energy density functionals based on Skyrme phenomenology and compare to thermodynamic equa- tions of state derived from chiral two- and three-nucleon forces in many-body perturbation theory. Quantum Monte Carlo simulations of symmetric nuclear matter and pure neutron matter are used to determine the density regimes in which perturbation theory with chiral nuclear forces is valid. Within the theoretical uncertainties associated with the many-body methods, we find that select mean field models describe well microscopic nuclear thermodynamics. As an additional consistency requirement, we study as well the single-particle properties of nucleons in a hot/dense environment, which affect e.g., charged-current weak reactions in neutron-rich matter. The identified mean field models can be used across a larger range of densities and temperatures in astrophysical simulations than more computationally expensive microscopic models.

Following the approach in the book "Commutative Algebra", by D. Eisenbud, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the generic initial extensor of a subset of a Grassmannian and then the double-generic initial ideal of a so-called GL-stable subset of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric properties of a Hilbert scheme: they provide a necessary condition for a Borel ideal to correspond to a point of a given irreducible component, lower bounds for the number of irreducible components in a Hilbert scheme and the maximal Hilbert function in every irreducible component. Moreover, we prove that every isolated component having a smooth double-generic initial ideal is rational. As a byproduct, we prove that the Cohen-Macaulay locus of the Hilbert scheme parameterizing subschemes of codimension 2 is the union of open subsets isomorphic to affine spaces. This improves results by J. Fogarty (1968) and R. Treger (1989)., Comment: 23 pages. Final version. The present version contains several changes, more precisely: correcting typos; a more complete Introduction, including new references to literature; improving the statements of Theorems 3.3 and 5.4 and adding details in Example 2.8 (numbering of this version)

We study the problem of an impurity in fully polarized (spin-up) low density neutron matter with the help of an accurate quantum Monte Carlo method in conjunction with a realistic nucleon-nucleon interaction derived from chiral effective field theory at next-to-next-to-leading-order. Our calculations show that the behavior of the proton spin-down impurity is very similar to that of a polaron in a fully polarized unitary Fermi gas. We show that our results can be used to put tight constraints on the time-odd parts of the energy density functional, independent of the time-even parts, in the density regime relevant to neutron-rich nuclei and compact astrophysical objects such as neutron stars and supernovae., Comment: 5 pages, 3 figures

We present fully non-perturbative quantum Monte Carlo calculations with non-local chiral effective field theory (EFT) interactions for the ground state properties of neutron matter. The equation of state, the nucleon chemical potentials and the momentum distribution in pure neutron matter up to one and a half times the nuclear saturation density are computed with a newly optimized chiral EFT interaction at next-to-next-to-leading order. This work opens the way to systematic order by order benchmarking of chiral EFT interactions, and \emph{ab initio} prediction of nuclear properties while respecting the symmetries of quantum chromodynamics., Comment: 5 pages, 4 figures, 1 table

We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $\mathrm L_H$ by applying the constructive methods of Gr\"obner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $\mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to find interesting topological properties, as for instance that $\mathrm L_H$ is connected and that its locus of radical liftings is open. Moreover, we show that every ideal defining an arithmetically Cohen-Macaulay scheme of codimension two has a radical lifting, giving in particular an answer to an open question posed by L. G. Roberts in 1989., Comment: the presentation of the results has been improved, new section (Section 6 of this version) concerning the torus action on the scheme of liftings, more detailed proofs in Section 7 of this version (Section 6 in the previous version), new example added (Example 8.5 of this version)

In this paper, we consider a monomial ideal J in P := A[x1,...,xn], over a commutative ring A, and we face the problem of the characterization for the family Mf(J) of all homogeneous ideals I in P such that the A-module P/I is free with basis given by the set of terms in the Groebner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes. For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Janet and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied for the special case of a strongly stable monomial ideal J. Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F(J), called stably complete, that allows an explicit description of the family Mf(J). We obtain stronger results if J is quasi stable, proving that F(J) is a Pommaret basis and Mf(J) has a natural structure of affine scheme. The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to., Comment: 22 pages

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $\mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that for all strongly stable ideals $J$, the marked schemes $\mathbf{Mf}_{J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes., Comment: final version, accepted for publication on Journal of Commutative Algebra

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of schemes with Hilbert function $u$. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting., Comment: 21 pages. Comments are welcome. More concise version with a slight change in the title. A further revised version has been accepted for publication in Experimental Mathematics

We introduce a novel many body method which combines two powerful many body techniques, viz., quantum Monte Carlo and coupled cluster theory. Coupled cluster wave functions are introduced as importance functions in a Monte Carlo method designed for the configuration interaction framework to provide rigorous upper bounds to the ground state energy. We benchmark our method on the homogeneous electron gas in momentum space. The importance function used is the coupled cluster doubles wave function. We show that the computational resources required in our method scale polynomially with system size. Our energy upper bounds are in very good agreement with previous calculations of similar accuracy, and they can be systematically improved by including higher order excitations in the coupled cluster wave function., Comment: Submitted to Physical Review Letters

We define marked sets and bases over a quasi-stable ideal $\mathfrak j$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of $\mathfrak j$ and a given integer $m$. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough $m$, is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results., Comment: 30 pages. Final version. In the present version Section 6 about flatness is improved, and new subsections concerning comparison with other existing computational methods (Section 7.1) and some complexity results (Section 7.2) were added