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Let us assume that $X$ is smooth and affine for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$. Reduce the equation moduloe the maximal ideal to get the reduced variety $\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty. If there is a smooth point $\bar P \in \bar X(\kappa)$ (and $\dim \bar X = \dim X$), then it lifts by Hensel's Lemma to a point on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of each point to points modulo the square of the maximal ideal. This comes down to looking at equations $f(\dots, a_i + \pi x_i, \dots)$, where $f$ runs through the equations defining $X$, $a_i$ are lifts of the coordinates of $P$ and $x_i$ are variables; divide the equations by the highest possible power of $\pi$, which is a generator of the maximal ideal, and reduce mod $\pi$. This gives equations for the reduction mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees that the process will terminate after finitely many lifting steps.

If $X$ is not smooth, consider its singular locus $X'$ first (recursively). If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous procedure on $X$. Since $X'(k)$ is empty, all points that lift indefinitely will end up on the smooth part of $X$, which again gives termination.

Magma has an implementation (IsLocallySoluble). See Nils Bruin, Some ternary Diophantine equations of signature (n, n, 2), In Bosma and Cannon, Discovering Mathematics with Magma, Springer-Verlag, Heidelberg, 2004.

Let us assume that $X$ is smooth and projective for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$. Reduce the equation moduloe the maximal ideal to get the reduced variety $\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty. If there is a smooth point $\bar P \in \bar X(\kappa)$ (and $\dim \bar X = \dim X$), then it lifts by Hensel's Lemma to a point on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of each point to points modulo the square of the maximal ideal. This comes down to looking at equations $f(\dots, a_i + \pi x_i, \dots)$, where $f$ runs through the equations defining an affine patch of $X$ containing a lift of $\bar P$, $a_i$ are lifts of the coordinates of $\bar P$ and $x_i$ are variables; divide the equations by the highest possible power of $\pi$, which is a generator of the maximal ideal, and reduce mod $\pi$. This gives equations for the reduction mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees that the process will terminate after finitely many lifting steps.

If $X$ is not smooth, consider its singular locus $X'$ first (recursively). If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous procedure on $X$. Since $X'(k)$ is empty, all points that lift indefinitely will end up on the smooth part of $X$, which again gives termination.

Magma has an implementation (IsLocallySoluble). See Nils Bruin, Some ternary Diophantine equations of signature (n, n, 2), In Bosma and Cannon, Discovering Mathematics with Magma, Springer-Verlag, Heidelberg, 2004.

Let us assume that $X$ is smooth and affine for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$. Reduce the equation moduloe the maximal ideal to get the reduced variety $\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty. If there is a smooth point $\bar P \in \bar X(\kappa)$ (and $\dim \bar X = \dim X$), then it lifts by Hensel's Lemma to a point on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of each point to points modulo the square of the maximal ideal. This comes down to looking at equations $f(\dots, a_i + \pi x_i, \dots)$, where $f$ runs through the equations defining $X$, $a_i$ are lifts of the coordinates of $P$ and $x_i$ are variables; divide the equations by the highest possible power of $\pi$, which is a generator of the maximal ideal, and reduce mod $\pi$. This gives equations for the reduction mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees that the process will terminate after finitely many lifting steps.

If $X$ is not smooth, consider its singular locus $X'$ first (recursively). If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous procedure on $X$. Since $X'(k)$ is empty, all points thet lift indefinitely will end up on the smooth part of $X$, which again gives termination.

Magma has an implementation (IsLocallySoluble). See Nils Bruin, Some ternary Diophantine equations of signature (n, n, 2), In Bosma and Cannon, Discovering Mathematics with Magma, Springer-Verlag, Heidelberg, 2004.

Let us assume that $X$ is smooth and affine for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$. Reduce the equation moduloe the maximal ideal to get the reduced variety $\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty. If there is a smooth point $\bar P \in \bar X(\kappa)$ (and $\dim \bar X = \dim X$), then it lifts by Hensel's Lemma to a point on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of each point to points modulo the square of the maximal ideal. This comes down to looking at equations $f(\dots, a_i + \pi x_i, \dots)$, where $f$ runs through the equations defining $X$, $a_i$ are lifts of the coordinates of $P$ and $x_i$ are variables; divide the equations by the highest possible power of $\pi$, which is a generator of the maximal ideal, and reduce mod $\pi$. This gives equations for the reduction mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees that the process will terminate after finitely many lifting steps.

If $X$ is not smooth, consider its singular locus $X'$ first (recursively). If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous procedure on $X$. Since $X'(k)$ is empty, all points that lift indefinitely will end up on the smooth part of $X$, which again gives termination.

Magma has an implementation (IsLocallySoluble). See Nils Bruin, Some ternary Diophantine equations of signature (n, n, 2), In Bosma and Cannon, Discovering Mathematics with Magma, Springer-Verlag, Heidelberg, 2004.

Let us assume that $X$ is smooth and affine for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$. Reduce the equation moduloe the maximal ideal to get the reduced variety $\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty. If there is a smooth point $\bar P \in \bar X(\kappa)$ (and $\dim \bar X = \dim x$), then it lifts by Hensel's Lemma to a point on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of each point to points modulo the square of the maximal ideal. This comes down to looking at equations $f(\dots, a_i + \pi x_i, \dots)$, where $f$ runs through the equations defining $X$, $a_i$ are lifts of the coordinates of $P$ and $x_i$ are variables; divide the equations by the highest possible power of $\pi$, which is a generator of the maximal ideal, and reduce mod $\pi$. This gives equations for the reduction mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees that the process will terminate after finitely many lifting steps.

If $X$ is not smooth, consider its singular locus $X'$ first (recursively). If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous procedure on $X$. Since $X'(k)$ is empty, all points thet lift indefinitely will end up on the smooth part of $X$, which again gives termination.

Magma has an implementation (IsLocallySoluble). See Nils Bruin, Some ternary Diophantine equations of signature (n, n, 2), In Bosma and Cannon, Discovering Mathematics with Magma, Springer-Verlag, Heidelberg, 2004.

Let us assume that $X$ is smooth and affine for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$. Reduce the equation moduloe the maximal ideal to get the reduced variety $\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty. If there is a smooth point $\bar P \in \bar X(\kappa)$ (and $\dim \bar X = \dim X$), then it lifts by Hensel's Lemma to a point on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of each point to points modulo the square of the maximal ideal. This comes down to looking at equations $f(\dots, a_i + \pi x_i, \dots)$, where $f$ runs through the equations defining $X$, $a_i$ are lifts of the coordinates of $P$ and $x_i$ are variables; divide the equations by the highest possible power of $\pi$, which is a generator of the maximal ideal, and reduce mod $\pi$. This gives equations for the reduction mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees that the process will terminate after finitely many lifting steps.

If $X$ is not smooth, consider its singular locus $X'$ first (recursively). If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous procedure on $X$. Since $X'(k)$ is empty, all points thet lift indefinitely will end up on the smooth part of $X$, which again gives termination.

Magma has an implementation (IsLocallySoluble). See Nils Bruin, Some ternary Diophantine equations of signature (n, n, 2), In Bosma and Cannon, Discovering Mathematics with Magma, Springer-Verlag, Heidelberg, 2004.