Tensor Product of Demazure Crystals

• Tensor Product of Demazure Crystals is a construction combining crystal bases from highest weight modules to reveal intricate algebraic structures.
• The approach leverages combinatorial and geometric techniques to analyze how tensoring affects crystal integrity and representation decompositions.
• Key insights include explicit tensor product decompositions, algorithmic methods for crystal operations, and applications to affine Lie algebra representations.

A robust variant of the Skolem, Positivity, and Ultimate Positivity Problems for linear recurrence sequences (LRS) involves determining whether a given recurrence remains strictly positive (or avoids zero) for all choices of initial configurations within a specified neighborhood of the nominal initial state. These robust variants lie between the classical, initialized versions and the "uninitialized" problems, and they occupy a significant frontier in the theory of linear dynamical systems and algorithmic number theory.

1. Definition of Robustness for Skolem and Positivity Problems

Given an order-$k$ LRS over $\mathbb{Q}$ or $\mathbb{R}$,

$$x_{n+k} = \sum_{i=0}^{k-1} a_i\,x_{n+i},\quad a_i\in\mathbb{Q}, \quad a_0\neq 0,$$

with initial vector $x^{(0)} = (x_0, \ldots, x_{k-1})\in\mathbb{R}^k$, questions of interest include:

• Classical Positivity: Is $x_n > 0$ for all $n\ge 0$?
• Robust Positivity: Given $\epsilon > 0$, does $x_n(x) > 0$ hold for all $x$ in the open $\epsilon$-ball $$N_\epsilon(x^{(0)}) = \{x: \|x - x^{(0)}\|_2 < \epsilon\}$$ and all $n\ge 0$?
• Robust Skolem Problem: Is $x_n(x)\neq 0$ for all $x\in N_\epsilon(x^{(0)})$, $n$?
• Existential Robustness: Does there exist $\epsilon > 0$ such that Robust Positivity (respectively Skolem) holds for all $x$ in $N_\epsilon(x^{(0)})$?
• Ultimate Positivity: Is there $N_0$ such that $x_n\ge 0$ for all $n\ge N_0$? There are further distinctions between "uniform" (the same $N_0$ works for all $x$ in a neighborhood) and "non-uniform" (an $N_x$ for each $x$).

2. Decidability and Complexity Landscape

When $\epsilon$ is Part of the Input (Fixed Ball):

Hardness Results:

• Robust Positivity and Robust Skolem are $\mathcal{T}$-Diophantine hard for LRS of order at least 6. Deciding these would allow effective approximation of Diophantine invariants $L(\theta)=\inf\{c: |\theta-n/m|< c/m^2\}$ for transcendental $\theta$—a central open problem in transcendence theory.
• Robust Uniform/Non-Uniform Ultimate Positivity (on closed balls) is $\mathcal{T}$-Lagrange hard, i.e., linked to the intractability of computing $L_\infty(\theta)=\liminf m |\theta - n/m| m^2$ (Akshay et al., 2022, Vahanwala, 2023).
• For the non-uniform, open-ball variant, the problem is in PSPACE: one can directly encode the requirement as a universal first-order formula in the theory of the reals and decide it algorithmically (Akshay et al., 2022, Vahanwala, 2023).

Existential Robustness ($\exists\epsilon>0$):

• Decidable in PSPACE: The existentially robust versions of Skolem and Positivity, as well as Uniform/Non-Uniform Ultimate Positivity, admit PSPACE algorithms. The key idea is to express the requirement for the existence of a robust neighborhood as a quantified formula in the first-order theory of the reals and apply quantifier elimination techniques (e.g., Renegar's algorithm) (Akshay et al., 2022, Vahanwala, 2023).

3. Structural and Geometric Techniques

Robust versions of these problems are naturally cast in terms of convex geometry:

• The set $$P = \{c \in \mathbb{R}^k \mid \forall n\ge 0, u_n(c)\ge 0\}$$ (the region of parameter space defining positive LRS) is convex, often a cone for the dominant part.
• To ensure robust positivity, the input ball $N_\epsilon(x^{(0)})$ must be fully contained in $P$.
• For ultimate positivity, the intersection with an appropriate family of half-spaces (representing large-$n$ constraints) defines a region that again can be checked with real algebraic methods (Akshay et al., 2022).
• Masser’s theorem is employed to find all integer relations among the dominant eigenangles $\theta_i$, yielding a compact torus $T$ that parameterizes the asymptotic behavior. Robust positivity reduces to checking that the image of the initial ball under all large powers of the recurrence operator remains in the positive cone, which is expressible as a quantifier statement in real algebraic geometry (Akshay et al., 2022).

4. Diophantine Hardness, Torus Geometry, and Quantifier Elimination

• At and beyond order 6, the robust positivity and Skolem problems encode fundamental questions of Diophantine approximation, making them at least as hard as distinguishing fine rational approximations to irrational numbers.
• For small orders or existential robustness, quantifier elimination (first-order theory of the reals—FO($\mathbb{R}$)) and algebraic geometry suffice. The complexity is controlled by the degree and number of quantified variables, but is generally in PSPACE.
• Key technical steps: use density properties of sequences on tori (Kronecker's theorem), explicit descriptions of positivity regions as intersections of half-spaces, and geometric reasoning about inclusion of balls in convex sets (Akshay et al., 2022, Vahanwala, 2023).

5. Variants: Uniform, Non-Uniform, Strict, and Ultimate Positivity

• Uniform Ultimate Positivity (robust): Does there exist a uniform $N$ such that for all $x$ within $N_\epsilon(x^{(0)})$ and all $n\ge N$, $x_n(x)\ge 0$? For fixed $\epsilon$, this variant is also generically hard (Lagrange hard); existential versions are in PSPACE.
• Non-uniform Ultimate Positivity (robust): For each $x$, does there exist $N_x$? On open balls, this is decidable in PSPACE even for explicit, fixed neighborhoods.
• Strict vs. Non-strict: One can define strict ($>0$) or weak ($\ge 0$) robust positivity, but the geometric and complexity properties are analogous; difference is in the algebraic inequalities checked (Akshay et al., 2022).
• Ultimate variants concern eventual rather than persistent positivity and admit similar but slightly more tractable complexity analyses.

6. Open Problems and Boundaries of Decidability

• Non-robust Skolem Problem: Decidability for general order remains open—robust variants inherit much of this complexity at higher orders.
• Complexity Sharpness: Determining whether existentially robust variants can be solved in better-than-PSPACE complexity remains unsettled.
• Rational Centers: Robustness centered at rational initial points (rather than arbitrary real ones) is not fully understood in terms of decidability and complexity.
• Extensions to Richer Dynamical Models: Generalizations to parameterized recurrences or non-linear dynamical systems pose significant technical challenges.
• Sharp characterization for closed ball, non-uniform ultimate positivity and higher-order cases demand significant number-theoretic advances (Akshay et al., 2022, Vahanwala, 2023).

7. Summary Table: Robust Positivity and Decidability

Variant	LRS Order	Neighborhood	Complexity/Hardness
Non-robust Positivity	$\le 5$	point	Decidable
Robust Positivity (fixed $\epsilon$)	$\ge 6$	open/closed ball	Diophantine-hard ($\mathcal{T}$-hard)
Robust Skolem (fixed $\epsilon$)	$\ge 6$	open/closed ball	Diophantine-hard ($\mathcal{T}$-hard)
Existential Robust (∃$\epsilon$)	any $k$ (order)	open/closed ball	Decidable in PSPACE
Robust Uniform Ultimate Positivity	$\ge 6$	closed ball	Lagrange-hard ($\mathcal{T}$-hard)
Robust Non-Uniform (open ball)	any $k$	open ball	Decidable in PSPACE

Decidability and hardness transition sharply at order 6 and are governed by Diophantine approximation problems (constants $L(\theta)$ and $L_\infty(\theta)$), tightly connecting dynamical systems theory, algebraic geometry, and deep transcendence questions in number theory (Akshay et al., 2022). The landscape is thus characterized by a pronounced phase transition: robust (and even classical) positivity becomes undecidable as soon as the analytic complexity of recurrence roots allows the encoding of hard Diophantine approximation instances.