@Giovanni: Low frequency means long wavelength is equally valid for linear polarized antennas i.e. horizontal or vertical polarization. If this is the case then why we prefer vertically polarized antennas over horizontal polarized at LF?

Your question cannot be answered correstly unless you precise your question:

- What frequency range do you consider 'low frequency'?

- What coverage area do you want to achieve? Around the broadcasting station? Or do you want to reach a specific area much further away?

Concerning high power (high power in broadcasting would be 100 kW to 2 MW):

The high voltage points of the antenna may cause lethal injuries when touched. In high humidity these point may arch over to the ground. Same in areas where bush fires (carbon particles) are frequent.

Electric field is the one we refer the polarisation to. Horizontal means that electric field and the related currents runs parallel to the ground, which is very lossy, and the losses are higher as your antenna is closer to the surface because of electric field coupling. At low frequencies your antenna is relatively close to the ground, and your losses are greater.

In case the surface is very conductive, such antenna's signal is effectively reflected upwards, as the antenna's image in such surface cancels it's emission at the horizon.

At several lambda above normal ground it makes little difference, except that the reflected path is less due to the same effect.

Verticals have less coupling with the ground and therefore less electric field related losses.

There is also a question of shortened antennas that theoretically may have the same gain as the full size dipoles, but seldom you'll see any calculation of losses at such structures. Trouble is that unless you have a beefy shortened structure, the material losses will make your antenna have real gain as low as it is shortened against the dipole. Situation goes worse when such short antenna is coupled with lossy ground.

For radio sound broadcasting in the upper HF broadcast bands (6 to 30 MHz) both horizontally and vertically polarized antennas will work, although horizontal polarization is more common. This is logical is, as these frequencies are generally used to reach far away (500-5000 km) areas by means of ionospheric radio wave propagation. This calls for directional antennas to beam the energy towards the coverage area, both in the horizontal and the vertical plane. Using horizontal polarization gives addition antenna gain due to the ground reflection near the antenna.

For radio sound broadcasting in the tropical bands (2 to 3 MHz) or in the lower shortwave bands (3 to 7 MHz) the antennas used may also be horizontally polarized antennas suspended at 0.1 to 0.2 lambda above ground. These antennas radiated straight upwards and that radiation is reflected by the ionosphere to cover a continuous area around the transmitter with a radius of 300-500 km (typically) with excellent signal strength. This propagation mechanism is called Near Vertical Incidence Skywave (NVIS).

In the higher medium wave (1 to 1.5 MHz) ionospheric radio wave propagation is still possible at nighttime and sufficiently high horizontally polarized antennas would work at night to cover longer distances. Vertical polarization may also be used for this. Daytime absorption in the ionosphere, however, is very high.

NVIS would be possible at nighttime, but suspending horizontal antennas much lower than 0.1 lambda above ground requires either an extensive ground screen or the amount of power lost into the ground below the antenna will be very high.

Ionospheric propagation is not possible or not practical at frequencies below 1 MHz. Therefore most broadcast station on long or medium wave (0.1 - 1.5 MHz) are designed for ground wave propagation, for which vertically polarized antennas work best. These antennas also need extensive ground screens, as the ground is part of the antenna itself and ground losses directly cause reduction of the antenna efficiency.

Antenna gain, ground losses and voltages between antenna and earth for horizontal and vertical antennas with `thin´ elements can be modeled quite well with free Method of Moments (MoM) antenna packages such as NEC-2 or NEC-4. Be sure to use Sommerfeld ground, and please compare your results with good antenna books (Kraus, Jasik) to avoid common simulation pitfalls. If your simulation are contradicting common theory, they might be wrong :-)

Direction of electric field is the direction of polarisation. In Horizontal polarisation,the   currents runs parallel to the ground, which is v. lossy, and thus losses are higher. The effect dominates at low frequency beacause the antenna is closer to ground The earth acts as a conductor at low freq and short the electric field, hence horizontyal polarisation is generally not used at low freq.

Because when antenna is horizontally polarized than EM waves propagate parallel to earth and at low frequencies earth becomes ionized and it behaves just like a conductor therefore  horizontally polarized  antenna is not preferred at low frequencies broadcast application

I recently found several artciles on NVIS antennas (that is: horizontal antennas radiating straight up towards the ionosphere) for medium wave broadcasting, so it can be done!

When doing so, it is important to read information about magnetoionic propagation, especially when you are lcoated close to the magnetic equator. You might want to adapt the antenna poalrization (antenna orientation) to the propagation. Davies, Ionospheric Radio, is a good book to start with.