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## Small Nonnegative Solutions of Additive Equations

### Diophantine Equations and Inequalities in Algebraic Number Fields (1991-01-01): 111-126 , January 01, 1991

Let α_{1}…,α_{2s} be 2*s* nonzero elements of *P.* Consider the additive equation of the type
9.1
$$
{\alpha _1}\lambda _1^k + \cdots + {\alpha _s}\lambda _s^k - {\alpha _{s + 1}}\lambda _{s + 1}^k - \cdots - {\alpha _{2s}}\lambda _{2s}^k = 0.
$$

## Periodic Functions

### Applications of Number Theory to Numerical Analysis (1981-01-01): 113-130 , January 01, 1981

The *G*_{s} may be regarded as tori. The 1-dimensional torus *G*_{1} may be obtained by identifying two end-points of the unit interval 0 ≤ *x*_{1} ≤ 1 and *G*_{2} by indentifying 2 opposite sides of the unit square 0 ≤ *x*_{1} ≤ 1, 0 ≤ *x*_{2} ≤ 1. In general, *G*_{s} is obtained by identifying the 2*s* opposite surfaces of the s-dimensional unit cube, i.e., the points
$$ \left( {{x_1},...,{x_{v - 1}},0,{x_{v + 1}},...,{x_s}} \right) $$
and
$$ \left( {{x_1},...,{x_{v - 1}},1,{x_{v + 1}},...,{x_s}} \right) $$
are identified, where 1 ≤ *v* ≤ *s*.

## Interpolation

### Applications of Number Theory to Numerical Analysis (1981-01-01): 183-203 , January 01, 1981

Let 1 < *n*^{1}<*n*_{2} ... be a sequence of integers and let
$$ {P_{{n_l}}}(k) = \left( {{x_1}^{\left( {{n^l}} \right)}(k),...,x_s^{\left( {{n_l}} \right)}(k)} \right),\quad 1 \leqslant k \leqslant {n_l},l = 1,2,... $$
be a sequence of uniformly distributed sets in *G*_{s}. For any given function *f*(*x*) on *G*_{s}, let
(9.1)
$$ {P_f}(x) = \sum\limits_{k = 1}^{{n_l}} f \left( {{P_{{n_l}}}(k)} \right){\varphi_{{n_l},k}}(x) $$
, where
$$ {\phi_{{{n_l},k}}}(x)\left( {1 \leqslant k \leqslant {n_1}} \right) $$
are given functions.

## Complete Exponential Sums

### Diophantine Equations and Inequalities in Algebraic Number Fields (1991-01-01): 14-22 , January 01, 1991

Let
$$
f\left( \lambda \right) = {{\alpha }_{k}}{{\lambda }^{k}} + ... + {{\alpha }_{1}}\lambda
$$
be a*k*-th degree polynomial with coefficients in *K*. Let a =(α_{k}…,α_{1}) be the fractional ideal generated by (α_{k}…,α_{1}). Suppose that aδ=g/q, where g;q are two relatively prime ideals, and
$$
S(f(x),q) = \sum\limits_{\lambda (q)} {E(f(\lambda ))} ,
$$

where λ runs over a complete residue system mod q.

## Chen Jingrun: A brief outline of his life and works

### Acta Mathematica Sinica (1996-09-01) 12: 225-233 , September 01, 1996

## Approximate Solution of Integral Equations and Differential Equations

### Applications of Number Theory to Numerical Analysis (1981-01-01): 204-223 , January 01, 1981

*If*
$$ \sum\limits_{i = 1}^s {\sum\limits_{j = 1}^s {{\alpha_{ij}}{x_i}{x_j}\left( {{\alpha_{ij}} = {\alpha_{ji}}} \right)} } $$
*is a semi-positive definite quadratic form, then*
$$ 0 \leqslant \det \left( {{\alpha_{ij}}} \right) \leqslant \prod\limits_{i = 1}^s {{\alpha_{ii}}} $$
.

## The Circle Method in Algebraic Number Fields

### Diophantine Equations and Inequalities in Algebraic Number Fields (1991-01-01): 58-71 , January 01, 1991

Let *h* and *t* be real numbers satisfying
$$
h > 2Dt, t > 1.
$$

## Recurrence Relations and Rational Approximation

### Applications of Number Theory to Numerical Analysis (1981-01-01): 28-47 , January 01, 1981

Let *F s* = Q(*α*) be a real algebraic number field of degree *s*. We shall give in this chapter an algorithm for the simultaneous Diophantine approximation obtained by *η*_{l} = *α*^{l} (*l* = 1, 2, ....) which is essentially the Jacobi-Perron algorithm (Cf. L. Bernstein [1]). It yields less precise results but the computations of *n*_{l} and *h*_{lj}.(1 ≤ *j* ≤ *s*) are comparatively simple.

## On small zeros of quadratic forms over finite fields (II)

### Acta Mathematica Sinica (1993-12-01) 9: 382-389 , December 01, 1993

Let
$$Q(\underline{\underline x} ) = Q(x_1 , \cdot \cdot \cdot x_n )$$
be a quadratic form with integer coefficients and let*p* denote a prime. Cochrane^{[1]} proved that if*n*≥4 then
$$Q(\underline{\underline x} ) = 0(\bmod p)$$
has a solution
$$\underline{\underline x} \ne \underline{\underline 0} $$
satisfying
$$\left| {\underline{\underline x} } \right| \ll \sqrt p $$
, where
$$\left| {\underline{\underline x} } \right| = \max \left| {x_i } \right|$$
. The aim of the present paper is to generalize the above result to finite fields.

## Additive Equations

### Diophantine Equations and Inequalities in Algebraic Number Fields (1991-01-01): 98-110 , January 01, 1991

Let α _{i} (1≤*i*≤*s*) be a set of nonzero integers. The form
$$A(\lambda ) = \sum\limits_{{i = 1}}^{s} {{{\alpha }_{i}}\lambda _{i}^{k}}$$
is called an *additive form*, and the equation
8.1
$$A(\lambda ) = 0$$
its corresponding *additive equation.* Let *R*_{s} (0) be the number of solutions of (8.1) subject to the condition: λ _{i} ∈ *P*(*T*), 1 ≤ *I* ≤ *s.* Then *R*_{s}(0) can be expressed as an integral over *U*_{n}; see §5.1. The corresponding singular series is
$$\mathfrak{S}(0) = \sum\limits_{\gamma } {G(\gamma ) = \sum\limits_{a} {H(\mathfrak{a}),} }$$
where
$$G(\gamma ) = \prod\limits_{{i = 1}}^{s} {{{G}_{i}}(\gamma ),}$$
$$\begin{array}{*{20}{c}} {{{G}_{i}}(\gamma ) = N{{{({{\mathfrak{a}}_{i}})}}^{{ - 1}}}\sum\limits_{{\lambda ({{\mathfrak{a}}_{i}})}} {E({{\alpha }_{i}}{{\lambda }^{k}}\gamma ),} } & {\gamma {{\alpha }_{i}}\delta \to {{\mathfrak{a}}_{i}},} & {1 \leqslant i \leqslant s,} \\ \end{array}$$
and where
$$H(\mathfrak{a}) = {{\mathop{\sum }\limits_{\gamma } }^{ \star }}G(\gamma )$$
in which γ runs over a reduced system of (αδ)^{-1} mod δ^{-1}.