arXiv subjects

Yihui Liang

Publications and source records attributed to Yihui Liang.

5 recordsLinked to original sources

arXiv2026

Optimized finite-$\beta$ tokamak-stellarator hybrid configurations achieved by planar dipole-field coils

Tokamak--stellarator hybrids seek to combine tokamak-like compactness and confinement with stellarator-like externally generated rotational transform and steady-state operation. In this work, we build on the recent tokamak--stellarator hybrid study using planar dipole-field coils (PDCs) [Yu et al., arXiv:2605.03599], in which the fixed-position, programmable coils on an axisymmetric winding surface generate flexible three-dimensional shaping fields. Using single-stage free-boundary optimization of coil currents and plasma-equilibrium parameters, we construct vacuum and finite-$\beta$ configurations. The vacuum cases show controllable external transform and magnetic well. The finite-$\beta$ cases accommodate various density, temperature, and pressure profiles, producing quasi-axisymmetric (QA) equilibria with self-consistent bootstrap current, favorable Mercier stability, and reduced demand for external current drive. Re-optimization enables $\beta$ ramp-up and access to different field-period QA branches with moderate coil-current changes. At large rotational transform, a toroidally omnigenous (TO)-like configuration exhibits more favorable infinite-$n$ ideal-ballooning behavior than a QA reference with matched profiles, even though ballooning stability is not directly optimized for. These results demonstrate that PDCs provide a flexible platform for achieving optimized finite-$\beta$ hybrid configurations.

physics.plasm-ph↗

arXiv2025

Explicit Stillman bounds for all degrees

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the length of an $R_\eta$-sequence containing fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result yields many other uniform bounds including bounds on the projective dimension of the ideals generated by $n$ forms of degree at most $d$. Explicit values of these bounds for forms of degree $5$ and higher are not yet known. This article constructs such explicit bounds, one of which is an upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by $n$ forms of degree at most $d$. In the settings of the Eisenbud-Goto conjecture, we derive an explicit bound of the Castelnuovo-Mumford regularity of a nondegenerate prime ideal $P$ in a polynomial ring $S$ in terms of the multiplicity of $S/P$.

math.AC↗

arXiv2022

Explicit Stillman bounds for all degrees

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by n forms of degree at most d. Explicit values of the bounds for forms of degrees 5 and higher are not yet known. The main result of this article is the construction of explicit such bounds, for all degrees d, which behave like power towers of height d^3/6+11d/6-4. This is done by establishing a bound D(k,d), which controls the number of generators of a minimal prime over an ideal of a regular sequence of k or fewer forms of degree d, and supplementing it into Ananyan and Hochster's proof in order to obtain a recurrence relation.

math.AC↗

arXiv2022

Upper bounds for regularity of radicals of ideals and arithmetic degrees

Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa an upper bound for the regularity of $\sqrt{I}$. More specifically we show that $\text{reg}(\sqrt{I})\leq d^{(n-1)2^{r-1}}$. In the second part, we show that the $r$-th arithmetic degree of $I$ is bounded above by $2\cdot d^{2^{n-r-1}}$. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.

math.AC↗

arXiv2020

Degree bounds for Gröbner bases of modules

Let $F$ be a non-negatively graded free module over a polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. Let $M$ be a submodule of $F$ generated by elements in $F$ with degrees bounded by $D$ and dim $F/M$=$r$. We prove that if $M$ is graded, the degree of the reduced Gröbner basis of $M$ for any term order is bounded by $2\left[1/2((Dm)^{n-r}m+D) \right]^{2^{r-1}}$. If $M$ is not graded, the bound is $2\left[1/2((Dm)^{(n-r)^2}m+D) \right]^{2^{r}}$. This is a generalization of Dubé(1990) and Mayr-Ritscher(2013)'s bounds for ideals in a polynomial ring.

math.AC↗