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Isotropic vectors over global fields

Przemysław Koprowski

TL;DR

This work provides a complete suite of algorithms for constructing isotropic vectors of quadratic forms over arbitrary global fields of characteristic not equal to $2$, covering all dimensions. The approach combines dimension-specific strategies: direct construction for dimensions $\le 3$, a quaternary ($4$-dimensional) isotropy test that reduces to solving a linear system over $\mathbb{F}_2$ in $S$-singular square classes, and higher-dimensional cases reduced step-by-step to lower dimensions via $2$-linkage and weak approximation. Central to the method are the notions of $S$-singular square classes $\mathbb{E}_S$, Hasse sets, and local-to-global criteria, guiding the construction of a suitable element $b$ (or $c$) that renders subforms isotropic; norm equations and Hilbert symbols underpin the correctness of the reductions. The algorithms are implemented in Magma and demonstrated to be practical through empirical timing analyses, with randomized strategies used where density arguments guarantee existence and convergence. Together, these results significantly broaden the algorithmic toolbox for quadratic forms over global fields and enable explicit isotropy vectors in diverse arithmetic settings.

Abstract

We present a complete suite of algorithms for finding isotropic vectors of quadratic forms (of any dimension) over an arbitrary global field of characteristic different from 2. This is a new version with numerous changes and improvements.

Isotropic vectors over global fields

TL;DR

This work provides a complete suite of algorithms for constructing isotropic vectors of quadratic forms over arbitrary global fields of characteristic not equal to , covering all dimensions. The approach combines dimension-specific strategies: direct construction for dimensions , a quaternary (-dimensional) isotropy test that reduces to solving a linear system over in -singular square classes, and higher-dimensional cases reduced step-by-step to lower dimensions via -linkage and weak approximation. Central to the method are the notions of -singular square classes , Hasse sets, and local-to-global criteria, guiding the construction of a suitable element (or ) that renders subforms isotropic; norm equations and Hilbert symbols underpin the correctness of the reductions. The algorithms are implemented in Magma and demonstrated to be practical through empirical timing analyses, with randomized strategies used where density arguments guarantee existence and convergence. Together, these results significantly broaden the algorithmic toolbox for quadratic forms over global fields and enable explicit isotropy vectors in diverse arithmetic settings.

Abstract

We present a complete suite of algorithms for finding isotropic vectors of quadratic forms (of any dimension) over an arbitrary global field of characteristic different from 2. This is a new version with numerous changes and improvements.
Paper Structure (10 sections, 5 theorems, 32 equations, 5 figures, 1 table)

This paper contains 10 sections, 5 theorems, 32 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Algorithmic prerequisites
  4. Algorithms
  5. Dimension 3 and below
  6. Dimension 4
  7. Dimension 5 and above
  8. Complexity of Algorithm \ref{['alg:dim4']}
  9. Experimental results
  10. Deterministic construction of elements $s$ and $t$

Key Result

Lemma 4.2

For every $\mathfrak{r}\in S_\infty$ the forms $\tilde{q}_{1}$ and $\tilde{q}_{2}$ are isotropic at $\mathfrak{r}$ if and only if

Figures (5)

  • Figure 1: Median running times of Algorithm \ref{['alg:dim4']} depending on the bit-size of the form. The thick gray line represents the total execution time. The dashed line is the time of the initialization phase. The thin solid line is the time spent in loop \ref{['st:dim4:loop']}. Finally, the dotted line shows the time used to solve norm equations in step \ref{['st:dim4:solve_ternary']}.
  • Figure 2: Median running times of Algorithm \ref{['alg:dim5_rnd']} depending on the bit-size of the form. The thick gray line represents the total execution time. The dashed line is the time of the initialization phase. The thin solid line is the time spent in loop \ref{['st:dim5rnd:loop']}. Finally, the dotted line shows the time used to find an isotropic vector of the $4$-dimensional form in step \ref{['st:dim5rnd:execute4']}.
  • Figure 3: Median running times of Algorithms \ref{['alg:dim4']} (top row) and \ref{['alg:dim5_rnd']} (bottom row) for number fields depending on the discriminant of the field. The thick gray line represents the total execution time. The dashed line is the time of the initialization phase. The thin solid line is the time spent in loop \ref{['st:dim4:loop']} of Algorithm \ref{['alg:dim4']} (top), or loop \ref{['st:dim5rnd:loop']} of Algorithm \ref{['alg:dim5_rnd']} (bottom). Finally, the dotted line shows the time used to find an isotropic vector of a lower-dimensional form.
  • Figure 4: Median running times of Algorithms \ref{['alg:dim4']} (left) and Algorithm \ref{['alg:dim5_rnd']} (right) depending on the degree of the base field. The thick gray line represents the total execution time. The dashed line is the time of the initialization phase. The thin solid line is the time spent in loop \ref{['st:dim4:loop']} of Algorithm \ref{['alg:dim4']} (left), or loop \ref{['st:dim5rnd:loop']} of Algorithm \ref{['alg:dim5_rnd']} (right). Finally, the dotted line shows the time used to find an isotropic vector of the lower-dimensional form.
  • Figure 5: Median running times of Algorithms \ref{['alg:dim4']} (left) and \ref{['alg:dim5_rnd']} (right) for rational function fields over finite fields depending on the size of the field of constants. The thick gray line represents the total execution time. The dashed line is the time of the initialization phase. The thin solid line is the time spent in loop \ref{['st:dim4:loop']} of Algorithm \ref{['alg:dim4']} (left), or loop \ref{['st:dim5rnd:loop']} of Algorithm \ref{['alg:dim5_rnd']} (right). Finally, the dotted line shows the time used to find an isotropic vector of the lower-dimensional form.

Theorems & Definitions (18)

  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Proposition 4.5
  • proof
  • Remark 1
  • Remark 2
  • ...and 8 more