Fixed-Precision p-Adic Representation

• Fixed-precision p-adic representation is defined by truncating the p-adic expansion at a power N, capturing both the known digits and the uncertainty using the notation x = p^v s + O(p^N).
• Arithmetic operations are carefully designed to propagate precision, with methods such as zealous, relaxed, and lattice-based approaches ensuring that precision loss is managed effectively.
• This representation is crucial in computational algebra systems and p-adic algorithmics, supporting applications from linear algebra problems to p-adic differential equations with optimal error monitoring.

A fixed-precision $p$-adic representation refers to the storage and computation of $p$-adic numbers with a prescribed level of truncation at a power $N$ of a prime $p$. In this model, a $p$-adic number $x$ is represented by its expansion as $x = \sum_{k=v}^{N-1} a_k p^k$, where each $a_k \in \{0,1,\ldots,p-1\}$, $v = \mathrm{val}_p(x)$ is the $p$-adic valuation (possibly negative), and $N$ is the precision bound, meaning only the digits from place $v$ up to $N-1$ are known, with the tail being undefined. This is notated concisely as $x = p^v s + O(p^N)$ for some integer $s$ coprime to $p$, encapsulating both the value and its uncertainty to the chosen fixed precision. The fixed-precision paradigm is foundational in $p$-adic algorithmics, appearing in both interval-based ("zealous") and more advanced lattice-based approaches, with extensive analysis in computational algebra systems and precision-propagation frameworks (Caruso, 2017, Caruso et al., 2018, Caruso et al., 2014).

1. Canonical Fixed-Precision $p$-Adic Expansion

The fixed-precision $p$-adic expansion is defined for a given $N \in \mathbb{Z}$ as the truncated expansion

$x = \sum_{k=v}^{N-1} a_k p^k$

with $a_k \in \{0,1,\ldots,p-1\}$, and is valid modulo $p^N$. This is equivalently written as

$$x = p^v s + O(p^N), \qquad 0 \leq s < p^{N-v},\quad \gcd(s,p)=1.$$

The value is exact up to $p^N$, and the remainder is unknown. Storage is typically either as $(v,N,s)$ triples (zealous paradigm) or as an integer mod $p^N$ (as in SageMath’s ZpL implementation), with underlying residue computation and explicit tracking of $p$-adic valuation and precision bounds (Caruso, 2017, Caruso et al., 2018).

2. Arithmetic Operations under Fixed Precision

Arithmetic with fixed-precision $p$-adic numbers proceeds with explicit attention to the propagation and loss of precision. Given two intervals $I = p^v s + O(p^N)$ and $J = p^{v'} s' + O(p^{N'})$:

• Addition:

$$I + J = p^{\min(v,v')}(s\,p^{v-\min(v,v')} + s'\,p^{v'-\min(v,v')}) + O(p^{\min(N,N')}),$$

with analogous formulae for subtraction.

• Multiplication:

$I \cdot J = a a' + O(p^{\min(v+N',v'+N)}),$

where the absolute precision is $\min(v+N', v'+N)$.

• Division (for $0 \not\in J$):

$I : J = a/a' + O(p^{\min(v+N'-2v', N-v')}),$

Efficient arithmetic is achieved via “schoolbook” algorithms or fast integer multiplication for large $N$. Inversion uses Newton iteration with $\tilde{O}(N)$ bit complexity per digit (Caruso, 2017).

3. Precision Management: Intervals versus Lattices

Precision in fixed-precision $p$-adics may be tracked at varying levels:

• Absolute precision: the smallest $k$ with residue known modulo $p^k$.
• Relative precision: $N-v$, the number of reliable digits.

While interval arithmetic tracks each variable independently, leading to pessimistic loss estimates, lattice-based precision maintenance encodes global linear relations among variables. This uses a $d\times d$ matrix (Hermite normal form over $\mathbb{Z}_p$) to capture the lattice $H \subset \mathbb{Q}_p^d$. The Caruso–Roe–Vaccon "Precision Lemma" asserts that for $f:U\subset E\rightarrow F$ with surjective differential $df_v$, the image of a small lattice $H$ under $f$ is precisely $f(v) + df_v(H)$. Thus, only true precision loss is modeled, avoiding artificial drops. ZpL, for example, implements this via automatic differentiation on $p$-adic operations, systematically updating a global precision lattice and capturing digit gains due to cancellation phenomena (Caruso, 2017, Caruso et al., 2014, Caruso et al., 2018).

4. Implementation Paradigms

Several paradigms exist for the implementation of fixed-precision $p$-adic arithmetic:

• Zealous arithmetic: Variables encapsulate $(v,N,s)$, guaranteeing correct inclusion but with pessimistic (over-estimated) loss of digits. Carries and operations are performed modulo $p^{N-v}$.
• Lazy/relaxed arithmetic: Operands are given as programs returning approximations mod $p^N$; arithmetic is performed only as needed for result precision. Relaxed methods (cf. van der Hoeven et al.) incrementally compute higher digits and enable fast integer multiplication via convolution and block tiling (Caruso, 2017).
• Floating-point $p$-adic arithmetic: Analogous to real floating-point, with values as $x \approx p^e\cdot s$ (with rounding). This is computationally efficient but lacks any formal guarantee on correctness of least significant digits.
• Lattice-based via automatic differentiation: Implementations (notably ZpL’s ZpLC and ZpLF) represent all live variables in a global precision lattice, updated differentially on every operation, with sharply minimal over-estimation (Caruso et al., 2018).

A direct comparative summary: | Paradigm | Guarantee | Precision loss | Computational cost | |-------------------|---------------|--------------------|-------------------------| | Zealous | Proven bounds | Often pessimistic | Costly for large objects| | Lazy/relaxed | Sharp bounds | Optimal | Higher memory/complexity| | Floating-point | None | Unmonitored | Fast, small constants | | Lattice-based | Sharp bounds | Optimal | Matrix/lattice updates |

5. Precision Propagation and Complexity Analysis

The theory of ultrametric precision, as applied in ZpL, uses automatic differentiation to propagate lattices optimally through sequences of operations. For an $n$-ary operation, propagating the precision lattice in ZpLF (arbitrary codimension) requires $O(nc_t)$ time, with $c_t$ the number of live variables (columns of the matrix); deletion similarly requires $O(\text{coind}_t(v) \cdot \text{cohgt}_t(v))$. Overall, if the underlying $p$-adic computation has cost $C$ and uses $s_{\rm in}$ inputs, the lattice-tracked cost in ZpLC is $O(s_{\rm in} C)$ (Caruso et al., 2018).

Precision updates for basic arithmetic, given scalars $x = a + O(p^{N_x})$, $y = b + O(p^{N_y})$, follow:

• Addition: $a + b + O(p^{\min(N_x,N_y)})$
• Multiplication: $$a b + O(p^{\min(N_x + \mathrm{val}_p(b), N_y + \mathrm{val}_p(a))})$$
• Inversion: $a^{-1} + O(p^{N_x - 2 \mathrm{val}_p(a)})$ These reflect the optimality of using Jacobian-based lattice updates and match the general formulae derived from first-order differentiation (Caruso et al., 2014).

6. Illustrative Applications and Numerical Behavior

Benchmarking and example computations demonstrate substantive differences between paradigms. For $p=2$, $N=10$, and linear algebra tasks (matrix LU, determinants, etc.):

• Zealous arithmetic loses $O(d)$ or $O(\sum \text{valuations})$ digits.
• Floating-point arithmetic preserves almost all digits, but their actual validity is not guaranteed.
• Relaxed and lattice-based approaches certify optimal digits.

In the computation of the unstable Somos-4 sequence, naive (stepwise) interval arithmetic quickly loses all precision, often failing to complete. Lattice-based tracking, however, leverages structural properties (e.g., the “Laurent phenomenon”) to maintain full digit count, as formalized in the propagation lemmas and illustrated via ZpL (Caruso, 2017, Caruso et al., 2014, Caruso et al., 2018).

For $p$-adic differential equations, e.g., repeated Newton iterations for power series over $\mathbb{Q}_2$, ZpL returns up to double the count of verified digits at high-order terms compared to standard approaches (Caruso et al., 2018).

7. Practical Guidelines and Implementation Recommendations

Practical fixed-precision $p$-adic types should consist of an integer residue $M$ mod $p^N$, together with exponent $N$. For composite structures (vectors, matrices, polynomials):

• Flat precision: single exponent $N$ across all entries;
• Jagged precision: per-coordinate exponents $(N_i)$;
• Full-lattice precision: $O$-module basis via integer matrix mod $p^N$.

Operators must update both residue and per-coordinate precision upon arithmetic. Whenever output precision falls below acceptable thresholds, error signaling or increased input precision is advised (Caruso et al., 2014). Optimization strategies recommend deferred precision management and lazy/relaxed computation to minimize unnecessary computation. For large objects, jagged precision with global offsets is preferred if correlations are provably non-contributive (Caruso et al., 2014).

Taken together, these methodologies enable black-box manipulation of $p$-adic numbers with precisely tracked, mathematically controlled precision, supporting robust computational workflows in number theory, arithmetic geometry, and related applications (Caruso et al., 2014, Caruso et al., 2018, Caruso, 2017).