Here is your question:
Is this planet plausible?
the planet is the third planet from its star.
by size smaller than Earth.
the planet has no moons.
the only habitable areas are rings around the poles.
in the pole, there's a sea of water.
ringed around the sea, is a habitable region with a tropical climate.
moving from the poles towards the equator, the land becomes more desert-like.
this planet does have seasons, and for the polar area that is its day and night cycle, each of which last about four and a half of our months.>
however, since the average temperature of this planet is mostly hot, no permanent snow nor ice caps occur.
the planet is the third planet from its star.
There are eight planets in our solar system. There are also known sytems of exoplanets around other stars.
As of 1 January 2022, there are 4,905 confirmed exoplanets in 3,629 planetary systems, with 808 systems having more than one planet.5
https://en.wikipedia.org/wiki/Exoplanet
With 1,279 excess planets in the 808 systems with more than one planet, there must be about 468 systems with three or more planets. Some systems have 7, 8 or maybe 9 planets. So it is perfectly possible for third planets to exist, and som of those third planets should be in the habitable zones of their stars.
2.
by size smaller than Earth.
3 planets in our solar system, and many exoplanets in other systems, are known to be smaller and/or less massive than Earth.
3.
the planet has no moons.
All the giant planets in our solar system have many moons of various sizes. The 4 rather EArth like terrestrial type planets include 2 with no moons, one with one moon, and one with 2 moons. Thus an Earthlike exoplanet without a moon seems perfectly possible. However, there are some theories claiming that having a large moon is vital for an explanet to be habitable, so you might want to check on those theories and see if you agree with them.
4.
the only habitable areas are rings around the poles.
In the outer solar system many minor planets and moons are covered with ice all over their surfaces, and are so cold that ice is as hard as rock or metal. Even their equators are far too cold for liquid water using life.
The surfaces of Mercury and Venus, close to the Sun, are far too hot for liquid water using life.
Earth, in between, has temperatures in most spots which are more or less good for liquid water using life. The temperatures are much warmer at the equator and much colder near the poles.
If an Earthlike planet was moved closer to its star, or the star became more luminous, its overall temperature would be higher, and the zones with temperatures best suitable for life would move away from the Equator and toward the poles.
If the warming process contiues, the zones with the best tmperatures for life would move farther and farther away from the poles, the equator would eventually become too hot for life, and eventually the very poles would be the only places cool enough for life, and then eventually even the very poles would be too hot for life.
So somewhere along the trend of increasing the heat received by the planet, it would have the right situation where only areas close to the poles would be habitable. All you have to do is decide how much radiation your world needs to receive from its star, and then adjust the luminosity of the star and the semi-major axis of the planet's orbit so it will receive that amount of radiation from its star.
If the polar oceans are surrounded by habitable land, the polar oceans will also be habitable, for sea life at least. Sea life is quite abundant in the Arctic and Southern Oceans, and on a hotter planet, the polar oceans could not be any colder.
5.
in the pole, there's a sea of water.
and:
6.
ringed around the sea, is a habitable region with a tropical climate.
The Arctic Ocean is almost surrouned by Eurasia, America, and Greenland. During glaciations, the strait between Siberia and Alaska becomes dry land, making the Arctic Ocean more enclosed.
With plate tectonics, ocean floor plates and continental plates move around the surface of Earh, and presumably on your planet. Thus continents and seas are constantly being rearranged over many millions of years.
Statistically, it would be rare for any specific planet to have polar oceans totally ringed in by encircling continents at any one time. But with constantly shifing continents and seas, a lot of planets might have such a configuration at least once while bearing life on their surfaces. And all the gazillions of planets in the universe, there are probably countless such planets existing at any one time.
You didn't mention whether there are any other oceans and continents on your world. There could be two polar oceans, two continents encircling them, and an equatorial ocean with several continents within it.
Or possibly there could be one continent coving the whole planet except for the two polar oceans it surrounds. In that case the two polar oceans would probably have to be much larger than the Arctic ocean, and extend much farther from the poles, if your planet has about the same amount of surface water as Earth has.
Possibly your planet is much dryer than Earth. If it recieves more radiation from a more luminous and/or hotter star, water in the equatorial regions would evaporate very fast. Water vapor which rose about the ozone layer of your world would be split up by the more intense ultraviolet radiation from the star into hydrogen and oxygen atoms, and the lighter hydroben atoms would escape into space. Thus your world could have tried up much faster than Earth and lost more water than Earth.
7.
moving from the poles towards the equator, the land becomes more desert-like.
If the regions closer to the equator are hotter, hot air will move toward the cooler polar regions. If the regions closer to the equator are hotter, water will evaporate faster and more water vapor will be carried to the poles and fall as rain there. Thus the lands closer to the equator will get less rain than the lands closer to the poles.
A desert is defined as a region with very little rain or snow, not as a hot region, the Antarctic and Greenland ice sheets are considered cold deserts. So the lands closer to the equator should be deserts or semi deserts.
What about the oceans close to the equator? I doubt whether oceans would ge considered deserts even if they didn't get any rain. So maybe your world as non desert regions in your equatorial oceans, or maybe the polar oceans are the only oceans. I discussed this under point 6.
8.
this planet does have seasons, and for the polar area that is its day and night cycle, each of which last about four and a half of our months.
You will have to decide how much axial tilt your planet will have.
At one extreme it could have a rotation axis perpendicular (at right angles) to the plane of its orbit, whichis called an axial tilt of zero. In that case it would have no seasons, merel night and day as it rotated.
On the other extreme, its axis of rotation could be exactly in the plane of its orbit, an axial tilt of 90 degrees.
At one point in the orbit, the north pole would be pointed directly at its star, and would have burning summer of constant less day, and the south pole would point directly away from the star, and have a freezing winter of constant night. Places on the equator would have the star costantly n thhorizon,a constant twilight/dawn,
Half a planetary year later the north pole would be pointed away from the star and have winter, and the south pole would b epoint toward the star and have summer.
And halfway between those two extremes, the equatorial planet of the planet would be pointing toward and away from the star. Each point on the equator would be having alternating days and nights.
You presumably will want a seasonal situation more like on Earth. Earth has an axial tilt of about 23 degrees, so I imagine you might consider making your planet have an axial tilt of about 20 or 25 degrees. Planets in our solar system have axial titls varying from 0.03 (Mercury) to 82.23 (Uranus), so that smaple gives you great freedom to choose.
But remember:
this planet does have seasons, and for the polar area that is its day and night cycle, each of which last about four and a half of our months.
So the polar area will have alternating day and night equal to their seasonal cycle. Day and night will last for half a planetary year in the polar regions. Since each day/summer season and each night/winter season last for about 4.5 Earht months, the year of your planet lasts for about 9 Earth months, or about 0.75 Earth years or about 273.9375 Earth days.
So you shuld find a star with a luminosity such that it has what I call an Earth Equivalent Distance or EED - where it receives about as much radiation from its star as Earth gets from the Sun - where the oribal period of a habitable planet woud be about 237 Earth days.
The answer by user177107 to this question:
https://astronomy.stackexchange.com/questions/40746/how-would-the-characteristics-of-a-habitable-planet-change-with-stars-of-differe/40758#40758
gives tables of some properties of various tpues of star,s including their EEDs, and the orbital periods at the EEDs of those stars.
According to the tables, a G8V class star would have an EED of 0.82 Astronimical uNits or AU, where the orbital period would be about 280.06 Earth days. So an 0.82 AU orbit about a G8V star would be good, except that you want the planet to be much hotter than Earth.
If a planet orbited the Sun at a distance if 0.84 AU, it would have an orbital period of about 281.153 Earth days. That would be farther from the Sun than Venus at 0.72 AU, but would that make it cool enough to have life even at the poles?
Just what are the inner and outer edges of the Sun's circumstellar habitable zone, when water on planetary surfaces can be liquid?
According to this list:
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates
A dozen different estimates, calcuations, and computer simulations have been made in the last 60 years, and some of them are very different from others.
Seven of them put the inner edge of the habitable zone over 0.9 AU from the Sun, while 4 or 5 others put it closer to the Sun than 0.84 AU. So you might might find it helpful to decide which inner edge of the habitable zone seems most correct to you.
And there are many other combinations of star mass, star luminosity, and semi-major axis of planetary orbit you could try in an attempt to find a situation where your world has a year about 9 months long and receives more radiation than Earth gets from the Sun.
yOu say the polar regions have day forhalf theyear followed by night for half the year. HOw far from the poles does that extend? 10 killometers, 100 kilometers, 1,000 kikometers? It that only in the centers of the polar oceans, or does it extend down to the shores of the habitable lands encircling the polar oceans?
On Earth, the sun can be seen at midnight on one day, and can be at the horizon at noon time on one other day, at any point north of the Arctic Circle or south of the Antarctic Circle. The farther beyond the circles, and the closer to the poles, one gets, the longer the periods of midnight sun and noonday darkness gets, until at the very poles the Sun is above or at the horizon for 6 months and then below the horizon for another 6 months.
And if you want the conditon of midnight sun/noonday darkness to extend down to a particular latitude on your world, you should figure out what axial tilt would be necessary - and if there is any axial tilt which can achieve it.
9.
however, since the average temperature of this planet is mostly hot, no permanent snow nor ice caps occur.
If you mean polar ice caps, the temperatures at the poles should be above freezing for longer than they are below, year after year after year, to prevent ice cap formation.
But what about glaciers on high monuntains? On Earth there are glaciers on high mountains and plateaus even near the poles. If your planet is hotter than Earth, even the highest mountains won't be able to have glaciers close to the Equator, but mountains in the temperate zones might have glaciers. If your planet is even hotter than that, even the highest mmountains in the temperate zones won't be able to have glaciers, but mountains in the polar regions might have glaciers. I fyour planet is hotter still, even the highest mmountains in the plolar regions won't be able to have glaciers.
But I suspect there might be deep valleys and pits somewhere near the poles where ice hidden from sunlight might be able to avoid melting and evaporating.
Scientists have detected ice on the surface of the Moon in deeply shadowed places at the poles.
https://www.nasa.gov/feature/ames/ice-confirmed-at-the-moon-s-poles
The planet Mercury gets very, very hot. Water ice has been detectdd in a crater at the north pole of Mercury.
https://en.wikipedia.org/wiki/Mercury_(planet)#Surface_conditions_and_exosphere
So I suppose that it is possible that some smal amounts of snow and ice might exist permanently in permenently shaded places near to the poles of your planet.
You don't want to make your planet so hot that no liquid water using life could live there. You might even want to make the polar regions cool enough to be habitable for humans.